Difference between revisions of "Trigonometric Integrals"
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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> | 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 | + | use the identities |
*<math>\sin(a)\cos(b)=\frac{\sin(a+b)+\sin(a-b)}{2}</math> | *<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>\sin(a)\sin(b)=\frac{\cos(a-b)-\cos(a+b)}{2}</math> | ||
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|<math>=\frac{\sin(x)}{2}-\frac{\sin(3x)}{6}+C</math> | |<math>=\frac{\sin(x)}{2}-\frac{\sin(3x)}{6}+C</math> | ||
|} | |} | ||
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==Resources== | ==Resources== |
Latest revision as of 17:34, 15 January 2022
When the integrand is primarily or exclusively based on trigonometric functions, the following techniques are useful.
Contents
Powers of Sine and Cosine
We will give a general method to solve generally integrands of the form . 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
This method works whenever there is an odd power of sine or cosine.
To evaluate when either or is odd.
- If is odd substitute and use the identity .
- If is odd substitute and use the identity .
Example
Find .
As there is an odd power of we let so . Notice that when we have and when we have .
When both and are even, things get a little more complicated.
To evaluate when both and are even.
Use the identities and .
Example
Find .
As and we have
and expanding, the integrand becomes
Using the multiple angle identities
TODO: CORRECT FORMULA
then we obtain on evaluating
Powers of Tan and Secant
To evaluate .
- If is even and then substitute and use the identity .
- If and are both odd then substitute and use the identity .
- If is odd and is even then use the identity and apply a reduction formula to integrate , using the examples below to integrate when .
Example 1
Find .
There is an even power of . Substituting gives so
Example 2
Find .
Let so . Then
Example 3
Find .
The trick to do this is to multiply and divide by the same thing like this:
Making the substitution so ,
More trigonometric combinations
For the integrals or or use the identities
Example 1
Find .
We can use the fact that , so
Now use the oddness property of to simplify
And now we can integrate
Example 2
Find: .
Using the identities
Then
Resources
- Trigonometric Integrals - Part 1 of 6 by patrickJMT
- Trigonometric Integrals - Part 2 of 6 by patrickJMT
- Trigonometric Integrals - Part 3 of 6 by patrickJMT
- Trigonometric Integrals - Part 4 of 6 by patrickJMT
- Trigonometric Integrals - Part 5 of 6 by patrickJMT
- Trigonometric Integrals - Part 6 of 6 by patrickJMT
- Trigonometric integrals - sin^mcos^n, odd m by Kriata King
- Trigonometric integrals - sin^mcos^n, odd n by Kriata King
- Trigonometric integrals - sin^mcos^n, m and n even by Kriata King
- Integrals of trigonometric functions, tan^msec^n, even n by Krista King
- Integrals of trigonometric functions, tan^msec^n, odd m by Krista King
Licensing
Content obtained and/or adapted from:
- Trigonometric integrals, Wikibooks: Calculus/Integration techniques under a CC BY-SA license