Difference between revisions of "Trigonometric Integrals"

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−
[https://youtu.be/Sv-bSbAQeXc Trigonometric Integrals Involving Powers of Sine and Cosine - Part 1] by James Sousa
 
  
−
[https://youtu.be/vPnuigP8I6I Trigonometric Integrals Involving Powers of Sine and Cosine - Part 2] by James Sousa
+
When the integrand is primarily or exclusively based on trigonometric functions, the following techniques are useful.
  
−
[https://youtu.be/RGaDMqhOg8Y Trigonometric Integrals Involving Powers of Secant and Tangent - Part 1] by James Sousa
+
===Powers of Sine and Cosine===
 +
We will give a general method to solve generally integrands of the form <math>\cos^m(x)\cdot\sin^n(x)</math> . First let us work through an example.
  
−
[https://youtu.be/o8sIHlS17qc Trigonometric Integrals Involving Powers of Secant and Tangent - Part 1] by James Sousa
+
:<math>\int\cos^3(x)\sin^2(x)dx</math>
  
−
[https://youtu.be/lTqnlihOC4o Trigonometric Integrals - Part 1 of 6] by patrickJMT
+
Notice that the integrand contains an odd power of cos. So rewrite it as
  
−
[https://youtu.be/zyg9k1je7Fg Trigonometric Integrals - Part 2 of 6] by patrickJMT
+
:<math>\int\cos^2(x)\sin^2(x)\cos(x)dx</math>
  
−
[https://youtu.be/BhJ4soojyAQ Trigonometric Integrals - Part 3 of 6] by patrickJMT
+
We can solve this by making the substitution <math>u=\sin(x)</math> so <math>du=\cos(x)dx</math> . Then we can write the whole integrand in terms of <math>u</math> by using the identity
 +
:<math>\cos^2(x)=1-\sin^2(x)=1-u^2</math> .
 +
So
 +
:{|
 +
|<math>\int\cos^3(x)\sin^2(x)dx</math>
 +
|<math>=\int\cos^2(x)\sin^2(x)\cos(x)dx</math>
 +
|-
 +
|
 +
|<math>=\int (1-u^2)u^2du</math>
 +
|-
 +
|
 +
|<math>=\int u^2du-\int u^4du</math>
 +
|-
 +
|
 +
|<math>=\frac{u^3}{3}+\frac{u^5}{5}+C</math>
 +
|-
 +
|
 +
|<math>=\frac{\sin^3(x)}{3}-\frac{\sin^5(x)}{5}+C</math>
 +
|}
  
−
[https://youtu.be/8CHHY-2Ctug Trigonometric Integrals - Part 4 of 6] by patrickJMT
+
This method works whenever there is an odd power of sine or cosine.
  
−
[https://youtu.be/QdNScjd5bno Trigonometric Integrals - Part 5 of 6] by patrickJMT
+
<blockquote style="background: white; border: 1px solid black; padding: 1em;">
 +
To evaluate <math>\int\cos^m(x)\sin^n(x)dx</math> when '''either''' <math>m</math> or <math>n</math> is '''odd'''.
 +
*If <math>m</math> is odd substitute <math>u=\sin(x)</math> and use the identity <math>\cos^2(x)=1-\sin^2(x)=1-u^2</math> .
 +
*If <math>n</math> is odd substitute <math>u=\cos(x)</math> and use the identity <math>\sin^2(x)=1-\cos^2(x)=1-u^2</math> .
 +
</blockquote>
  
−
[https://youtu.be/m3zG7c52QR4 Trigonometric Integrals - Part 6 of 6] by patrickJMT
+
====Example====
 +
Find <math>\int\limits_0^\frac{\pi}{2} \cos^{40}(x)\sin^3(x)dx</math> .
  
−
[https://youtu.be/WYhyq_mTCZs Trigonometric integrals - sin^mcos^n, odd m] by Kriata King
+
As there is an odd power of <math>\sin</math> we let <math>u=\cos(x)</math> so <math>du=-\sin(x)dx</math> . Notice that when <math>x=0</math> we have <math>u=\cos(0)=1</math> and when <math>x=\frac{\pi}{2}</math> we have <math>u=\cos(\tfrac{\pi}{2})=0</math> .
  
−
[https://youtu.be/RRDiT-djQPk Trigonometric integrals - sin^mcos^n, odd n] by Kriata King
+
:{|
 +
|<math>\int\limits_0^\frac{\pi}{2} \cos^{40}(x)\sin^3(x)dx</math>
 +
|<math>=\int\limits_0^\frac{\pi}{2} \cos^{40}(x)\sin^2(x)\sin(x)dx</math>
 +
|-
 +
|
 +
|<math>=-\int\limits_1^0 u^{40}(1-u^2)du</math>
 +
|-
 +
|
 +
|<math>=\int\limits_0^1 u^{40}(2-u^2)du</math>
 +
|-
 +
|
 +
|<math>=\int\limits_0^5
 +
(u^{40}-u^{42})du</math>
 +
|-
 +
|
 +
|<math>=\left(\frac{u^{41}}{41}-\frac{u^{43}}{43}\right)\Bigg|_0^1</math>
 +
|-
 +
|
 +
|<math>=\frac{1}{41}-\frac{1}{43}</math>
 +
|}
  
−
[https://youtu.be/rpbr2nH7lNY Trigonometric integrals - sin^mcos^n, m and n even] by Kriata King
+
When both <math>m</math> and <math>n</math> are even, things get a little more complicated.
  
−
[https://youtu.be/CK2SfrzF_c4 Integrals of trigonometric functions, tan^msec^n, even n] by Krista King
+
<blockquote style="background: white; border: 1px solid black; padding: 1em;">
 +
To evaluate <math>\int\cos^m(x)\sin^n(x)dx</math> when both <math>m</math> and <math>n</math> are '''even'''.
  
−
[https://youtu.be/0TuZSSah5hc Integrals of trigonometric functions, tan^msec^n, odd m] by Krista King
+
<br>Use the identities <math>\sin^2(x)=\frac{1-\cos(2x)}{2}</math> and <math>\cos^2(x)=\frac{1+\cos(2x)}{2}</math> .
 +
</blockquote>
 +
 
 +
====Example====
 +
Find <math>\int\sin^2(x)\cos^4(x)dx</math> .
 +
 
 +
As <math>\sin^2(x)=\frac{1-\cos(2x)}{2}</math> and <math>\cos^2(x)=\frac{1+\cos(2x)}{2}</math> we have
 +
:<math>\int\sin^2(x)\cos^4(x)dx=\int\left(\frac{1-\cos(2x)}{2}\right)\left(\frac{1+\cos(2x)}{2}\right)^2dx</math>
 +
and expanding, the integrand becomes
 +
:<math>\frac{1}{8}\int\left(1-\cos^2(2x)+\cos(2x)-\cos^3(2x)\right)dx</math>
 +
 
 +
Using the multiple angle identities
 +
 
 +
:{|
 +
|<math>I</math>
 +
|<math>=\frac{1}{8}\left(\int 1dx-\int\cos^2(2x)dx+\int\cos(2x)dx-\int\cos^3(2x)dx\right)</math>
 +
|-
 +
|
 +
|<math>=\frac{1}{8}\left(x-\frac{1}{2}\int\Big(1+\cos(4x)\Big)dx+\frac{\sin(2x)}{2}-\int\cos^2(2x)\cos(2x)dx\right)</math>
 +
|-
 +
|
 +
|TODO: CORRECT FORMULA<math>=\frac{1}{164}\left(x+\sin(2x)+\int\cos(4x)dx-2\int\Big(1-\sin^2(2x)\Big)\cos(2x)dx\right)</math>
 +
|}
 +
 
 +
then we obtain on evaluating
 +
:<math>I=\frac{x}{16}-\frac{\sin(4x)}{64}+\frac{\sin^3(2x)}{48}+C</math>
 +
 
 +
===Powers of Tan and Secant===
 +
<blockquote style="background: white; border: 1px solid black; padding: 1em;">
 +
To evaluate <math>\int\tan^m(x)\sec^n(x)dx</math> .
 +
#If <math>n</math> is even and <math>n\ge 2</math> then  substitute <math>u=tan(x)</math> and use the identity <math>\sec^2(x)=1+\tan^2(x)</math> .
 +
#If <math>n</math> and <math>m</math> are both odd then substitute <math>u=\sec(x)</math> and use the identity <math>\tan^2(x)=\sec^2(x)-1</math> .
 +
#If <math>n</math> is odd and <math>m</math> is even then use the identity <math>\tan^2(x)=\sec^2(x)-1</math> and apply a reduction formula to integrate <math>\sec^j(x)dx</math> , using the examples below to integrate when <math>j=1,2</math> .
 +
</blockquote>
 +
 
 +
====Example 1====
 +
Find <math>\int\sec^2(x)dx</math> .
 +
 
 +
There is an even power of <math>\sec(x)</math> . Substituting <math>u=\tan(x)</math> gives <math>du=\sec^2(x)dx</math> so
 +
 
 +
<math>\int\sec^2(x)dx=\int du=u+C=\tan(x)+C.</math>
 +
 
 +
 
 +
====Example 2====
 +
Find <math>\int\tan(x)dx</math> .
 +
 
 +
Let <math>u=\cos(x)</math> so <math>du=-\sin(x)dx</math> . Then
 +
 
 +
:{|
 +
|<math>\int\tan(x)dx</math>
 +
|<math>=\int\frac{\sin(x)}{\cos(x)}dx</math>
 +
|-
 +
|
 +
|<math>=\int -\frac{du}{u}</math>
 +
|-
 +
|
 +
|<math>=-\ln|u|+C</math>
 +
|-
 +
|
 +
|<math>=-\ln\Big|\cos(x)\Big|+C</math>
 +
|-
 +
|
 +
|<math>=\ln\Big|\sec(x)\Big|+C</math>
 +
|}
 +
 
 +
 
 +
====Example 3====
 +
Find <math>\int\sec(x)dx</math> .
 +
 
 +
The trick to do this is to multiply and divide by the same thing like this:
 +
 
 +
:{|
 +
|<math>\int\sec(x)dx</math>
 +
|<math>=\int\sec(x)\frac{\sec(x)+\tan(x)}{\sec(x)+\tan(x)}dx</math>
 +
|-
 +
|
 +
|<math>=\int\frac{\sec^2(x)+\sec(x)\tan(x)}{\sec(x)+\tan(x)}dx</math>
 +
|}
 +
 
 +
Making the substitution <math>u=\sec(x)+\tan(x)</math> so <math>du=\sec(x)\tan(x)+\sec^2(x)dx</math> ,
 +
 
 +
:{|
 +
|<math>\int\sec(x)dx</math>
 +
|<math>=\int\frac{du}{u}</math>
 +
|-
 +
|
 +
|<math>=\ln|u|+C</math>
 +
|-
 +
|
 +
|<math>\ln\Big|\sec(x)+\tan(x)\Big|+C</math>
 +
|}
 +
 
 +
===More trigonometric combinations===
 +
<blockquote style="background: white; border: 1px solid black; padding: 1em;">
 +
For the integrals <math>\int\sin(nx)\cos(mx)dx</math> or <math>\int\sin(nx)\sin(mx)dx</math> or <math>\int\cos(nx)\cos(mx)dx</math>
 +
use the [[Calculus/Table_of_Trigonometry|identities]]
 +
*<math>\sin(a)\cos(b)=\frac{\sin(a+b)+\sin(a-b)}{2}</math>
 +
*<math>\sin(a)\sin(b)=\frac{\cos(a-b)-\cos(a+b)}{2}</math>
 +
*<math>\cos(a)\cos(b)=\frac{\cos(a-b)+\cos(a+b)}{2}</math>
 +
</blockquote>
 +
 
 +
====Example 1====
 +
Find <math>\int\sin(3x)\cos(5x)dx</math> .
 +
 
 +
We can use the fact that <math>\sin(a)\cos(b)=\frac{\sin(a+b)+\sin(a-b)}{2}</math> , so
 +
:<math>\sin(3x)\cos(5x)=\frac{\sin(8x)+\sin(-2x)}{2}</math>
 +
Now use the oddness property of <math>\sin(x)</math> to simplify
 +
:<math>\sin(3x)\cos(5x)=\frac{\sin(8x)-\sin(2x)}{2}</math>
 +
And now we can integrate
 +
:{|
 +
|<math>\int\sin(3x)\cos(5x)dx</math>
 +
|<math>=\int\Big(\frac{\sin(8x)-\sin(2x)}{2}\Big)dx</math>
 +
|-
 +
|
 +
|<math>=\frac{\cos(2x)}{4}-\frac{\cos(8x)}{16}+C</math>
 +
|}
 +
 
 +
====Example 2====
 +
Find:<math>\int\sin(x)\sin(2x)dx</math> .
 +
 
 +
Using the identities
 +
:<math>\sin(x)\sin(2x)=\frac{\cos(-x)-\cos(3x)}{2}=\frac{\cos(x)-\cos(3x)}{2}</math>
 +
Then
 +
:{|
 +
|<math>\int\sin(x)\sin(2x)dx</math>
 +
|<math>=\frac{1}{2}\int\Big(\cos(x)-\cos(3x)\Big)dx</math>
 +
|-
 +
|
 +
|<math>=\frac{\sin(x)}{2}-\frac{\sin(3x)}{6}+C</math>
 +
|}
 +
 
 +
 
 +
 
 +
==Resources==
 +
*[https://youtu.be/Sv-bSbAQeXc Trigonometric Integrals Involving Powers of Sine and Cosine - Part 1] by James Sousa
 +
 
 +
*[https://youtu.be/vPnuigP8I6I Trigonometric Integrals Involving Powers of Sine and Cosine - Part 2] by James Sousa
 +
 
 +
*[https://youtu.be/RGaDMqhOg8Y Trigonometric Integrals Involving Powers of Secant and Tangent - Part 1] by James Sousa
 +
 
 +
*[https://youtu.be/o8sIHlS17qc Trigonometric Integrals Involving Powers of Secant and Tangent - Part 1] by James Sousa
 +
 
 +
*[https://youtu.be/lTqnlihOC4o Trigonometric Integrals - Part 1 of 6] by patrickJMT
 +
 
 +
*[https://youtu.be/zyg9k1je7Fg Trigonometric Integrals - Part 2 of 6] by patrickJMT
 +
 
 +
*[https://youtu.be/BhJ4soojyAQ Trigonometric Integrals - Part 3 of 6] by patrickJMT
 +
 
 +
*[https://youtu.be/8CHHY-2Ctug Trigonometric Integrals - Part 4 of 6] by patrickJMT
 +
 
 +
*[https://youtu.be/QdNScjd5bno Trigonometric Integrals - Part 5 of 6] by patrickJMT
 +
 
 +
*[https://youtu.be/m3zG7c52QR4 Trigonometric Integrals - Part 6 of 6] by patrickJMT
 +
 
 +
*[https://youtu.be/WYhyq_mTCZs Trigonometric integrals - sin^mcos^n, odd m] by Kriata King
 +
 
 +
*[https://youtu.be/RRDiT-djQPk Trigonometric integrals - sin^mcos^n, odd n] by Kriata King
 +
 
 +
*[https://youtu.be/rpbr2nH7lNY Trigonometric integrals - sin^mcos^n, m and n even] by Kriata King
 +
 
 +
*[https://youtu.be/CK2SfrzF_c4 Integrals of trigonometric functions, tan^msec^n, even n] by Krista King
 +
 
 +
*[https://youtu.be/0TuZSSah5hc Integrals of trigonometric functions, tan^msec^n, odd m] by Krista King

Revision as of 00:57, 9 October 2021

When the integrand is primarily or exclusively based on trigonometric functions, the following techniques are useful.

Powers of Sine and Cosine

We will give a general method to solve generally integrands of the form Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos^m(x)\cdot\sin^n(x)} . First let us work through an example.

Notice that the integrand contains an odd power of cos. So rewrite it as

We can solve this by making the substitution so . Then we can write the whole integrand in terms of by using the identity

.

So

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\int (1-u^2)u^2du}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\int u^2du-\int u^4du}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\frac{u^3}{3}+\frac{u^5}{5}+C}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\frac{\sin^3(x)}{3}-\frac{\sin^5(x)}{5}+C}

This method works whenever there is an odd power of sine or cosine.

To evaluate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\cos^m(x)\sin^n(x)dx} when either Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} is odd.

  • If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} is odd substitute Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u=\sin(x)} and use the identity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos^2(x)=1-\sin^2(x)=1-u^2} .
  • If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} is odd substitute Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u=\cos(x)} and use the identity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin^2(x)=1-\cos^2(x)=1-u^2} .

Example

Find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\limits_0^\frac{\pi}{2} \cos^{40}(x)\sin^3(x)dx} .

As there is an odd power of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin} we let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u=\cos(x)} so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle du=-\sin(x)dx} . Notice that when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=0} we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u=\cos(0)=1} and when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=\frac{\pi}{2}} we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u=\cos(\tfrac{\pi}{2})=0} .

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\limits_0^\frac{\pi}{2} \cos^{40}(x)\sin^3(x)dx} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\int\limits_0^\frac{\pi}{2} \cos^{40}(x)\sin^2(x)\sin(x)dx}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =-\int\limits_1^0 u^{40}(1-u^2)du}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\int\limits_0^1 u^{40}(2-u^2)du}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\int\limits_0^5 (u^{40}-u^{42})du}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\left(\frac{u^{41}}{41}-\frac{u^{43}}{43}\right)\Bigg|_0^1}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\frac{1}{41}-\frac{1}{43}}

When both Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} are even, things get a little more complicated.

To evaluate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\cos^m(x)\sin^n(x)dx} when both Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} are even.


Use the identities Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin^2(x)=\frac{1-\cos(2x)}{2}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos^2(x)=\frac{1+\cos(2x)}{2}} .

Example

Find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\sin^2(x)\cos^4(x)dx} .

As Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin^2(x)=\frac{1-\cos(2x)}{2}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos^2(x)=\frac{1+\cos(2x)}{2}} we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\sin^2(x)\cos^4(x)dx=\int\left(\frac{1-\cos(2x)}{2}\right)\left(\frac{1+\cos(2x)}{2}\right)^2dx}

and expanding, the integrand becomes

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{8}\int\left(1-\cos^2(2x)+\cos(2x)-\cos^3(2x)\right)dx}

Using the multiple angle identities

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\frac{1}{8}\left(\int 1dx-\int\cos^2(2x)dx+\int\cos(2x)dx-\int\cos^3(2x)dx\right)}
TODO: CORRECT FORMULA

then we obtain on evaluating

Powers of Tan and Secant

To evaluate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\tan^m(x)\sec^n(x)dx} .

  1. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} is even and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\ge 2} then substitute Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u=tan(x)} and use the identity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sec^2(x)=1+\tan^2(x)} .
  2. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} are both odd then substitute Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u=\sec(x)} and use the identity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tan^2(x)=\sec^2(x)-1} .
  3. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} is odd and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} is even then use the identity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tan^2(x)=\sec^2(x)-1} and apply a reduction formula to integrate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sec^j(x)dx} , using the examples below to integrate when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j=1,2} .

Example 1

Find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\sec^2(x)dx} .

There is an even power of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sec(x)} . Substituting Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u=\tan(x)} gives Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle du=\sec^2(x)dx} so

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\sec^2(x)dx=\int du=u+C=\tan(x)+C.}


Example 2

Find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\tan(x)dx} .

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u=\cos(x)} so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle du=-\sin(x)dx} . Then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\tan(x)dx} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\int\frac{\sin(x)}{\cos(x)}dx}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\int -\frac{du}{u}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =-\ln|u|+C}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =-\ln\Big|\cos(x)\Big|+C}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\ln\Big|\sec(x)\Big|+C}


Example 3

Find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\sec(x)dx} .

The trick to do this is to multiply and divide by the same thing like this:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\sec(x)dx} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\int\sec(x)\frac{\sec(x)+\tan(x)}{\sec(x)+\tan(x)}dx}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\int\frac{\sec^2(x)+\sec(x)\tan(x)}{\sec(x)+\tan(x)}dx}

Making the substitution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u=\sec(x)+\tan(x)} so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle du=\sec(x)\tan(x)+\sec^2(x)dx} ,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\sec(x)dx} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\int\frac{du}{u}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\ln|u|+C}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ln\Big|\sec(x)+\tan(x)\Big|+C}

More trigonometric combinations

For the integrals Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\sin(nx)\cos(mx)dx} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\sin(nx)\sin(mx)dx} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\cos(nx)\cos(mx)dx} use the identities

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(a)\cos(b)=\frac{\sin(a+b)+\sin(a-b)}{2}}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(a)\sin(b)=\frac{\cos(a-b)-\cos(a+b)}{2}}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos(a)\cos(b)=\frac{\cos(a-b)+\cos(a+b)}{2}}

Example 1

Find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\sin(3x)\cos(5x)dx} .

We can use the fact that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(a)\cos(b)=\frac{\sin(a+b)+\sin(a-b)}{2}} , so

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(3x)\cos(5x)=\frac{\sin(8x)+\sin(-2x)}{2}}

Now use the oddness property of to simplify

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(3x)\cos(5x)=\frac{\sin(8x)-\sin(2x)}{2}}

And now we can integrate

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\sin(3x)\cos(5x)dx} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\int\Big(\frac{\sin(8x)-\sin(2x)}{2}\Big)dx}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\frac{\cos(2x)}{4}-\frac{\cos(8x)}{16}+C}

Example 2

Find:Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\sin(x)\sin(2x)dx} .

Using the identities

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(x)\sin(2x)=\frac{\cos(-x)-\cos(3x)}{2}=\frac{\cos(x)-\cos(3x)}{2}}

Then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\sin(x)\sin(2x)dx} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\frac{1}{2}\int\Big(\cos(x)-\cos(3x)\Big)dx}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\frac{\sin(x)}{2}-\frac{\sin(3x)}{6}+C}


Resources

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