Difference between revisions of "Integrals Resulting in Inverse Trigonometric Functions"
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− | <math> \int\ | + | <math> \int\frac{du}{\sqrt{a^2 - u^2}} = \arcsin \left(\frac{u}{a}\right) + C </math> |
− | + | <math> \int\frac{du}{a^2+u^2} =\dfrac{1}{a}\arctan \left(\dfrac{u}{a}\right) + C </math> | |
− | [https://youtu.be/MdsAvt9y5ds Integrating using Inverse Trigonometric Functions] by patrickJMT | + | <math> \int\frac{du}{u\sqrt{u^2 - a^2}} =\frac{1}{a}\arcsec \left(\dfrac{|u|}{a}\right) + C </math> |
+ | |||
+ | |||
+ | ===Example 1=== | ||
+ | <p>Evaluate the integral</p> | ||
+ | |||
+ | <p class="mt-align-center"><math> \int\dfrac{dx}{\sqrt{4 - 9x^2}}.</math></p> | ||
+ | |||
+ | <p><strong>Solution</strong></p> | ||
+ | |||
+ | <p>Substitute <math> u=3x</math>. Then <math> du=3dx</math> and we have</p> | ||
+ | |||
+ | <p style="text-align: center;"><math> \int\dfrac{dx}{\sqrt{4 - 9x^2}}=\dfrac{1}{3}\int\dfrac{du}{\sqrt{4 - u^2}}.</math></p> | ||
+ | |||
+ | <p>Applying the formula with <math> a=2, </math> we obtain</p> | ||
+ | |||
+ | <p class="mt-indent-3" style="text-align:center;"><math> \int\dfrac{dx}{\sqrt{4 - 9x^2}}=\dfrac{1}{3}\int\dfrac{du}{\sqrt{4 - u^2}}=\dfrac{1}{3}\arcsin \left(\dfrac{u}{2}\right)+C=\dfrac{1}{3}\arcsin \left(\dfrac{3x}{2}\right)+C.</math></p> | ||
+ | |||
+ | ===Example 2=== | ||
+ | <p>Evaluate <math> \int \frac{4-x}{\sqrt{16-x^2}}\text{dx} </math>.</p> | ||
+ | |||
+ | <p><strong>Solution</strong></p> | ||
+ | |||
+ | <p>This integral requires two different methods to evaluate it. We get to those methods by splitting up the integral:</p> | ||
+ | |||
+ | <p><math> \int \frac{4-x}{\sqrt{16-x^2}}\text{dx} = \int \frac{4}{\sqrt{16-x^2}}\text{dx} - \int \frac{x}{\sqrt{16-x^2}}\text{dx} </math> </p> | ||
+ | |||
+ | <p>The first integral is handled straightforward; the second integral is handled by substitution, with <math>u = 16-x^2</math>. We handle each separately. </p> | ||
+ | |||
+ | <p><math>\int \frac{4}{\sqrt{16-x^2}}\text{dx} = 4\arcsin\frac{x}{4} + C.</math></p> | ||
+ | |||
+ | <p><math>\int\frac{x}{\sqrt{16-x^2}}\text{dx} </math>: Set <math>u = 16-x^2</math>, so <math>\text{du} = -2x\text{dx} </math> and <math>x\text{dx} = -\text{du} /2</math>. We have</p> | ||
+ | |||
+ | <p><math>\begin{align} \int\frac{x}{\sqrt{16-x^2}}\text{dx} = \int\frac{-\text{du} /2}{\sqrt{u}}\\ = -\frac12\int \frac{1}{\sqrt{u}}\text{du} \\ = - \sqrt{u} + C\\ = -\sqrt{16-x^2} + C.\end{align}</math></p> | ||
+ | |||
+ | <p>Combining these together, we have</p> | ||
+ | |||
+ | <p><math> \int \frac{4-x}{\sqrt{16-x^2}}\text{dx} = 4\arcsin\frac x4 + \sqrt{16-x^2}+C.</math></p> | ||
+ | |||
+ | ==Resources== | ||
+ | *[https://youtu.be/AE-0gXXx_j0 Integration into Inverse trigonometric functions using Substitution] by The Organic Chemistry Tutor | ||
+ | |||
+ | *[https://youtu.be/MdsAvt9y5ds Integrating using Inverse Trigonometric Functions] by patrickJMT | ||
+ | |||
+ | ==Licensing== | ||
+ | Content obtained and/or adapted from: | ||
+ | * [https://math.libretexts.org/Courses/Monroe_Community_College/MTH_211_Calculus_II/Chapter_5%3A_Integration/5.7%3A_Integrals_Resulting_in_Inverse_Trigonometric_Functions_and_Related_Integration_Techniques Integrals Resulting in Inverse Trigonometric Functions, LibreTexts: Mathematics] under a CC BY-SA-NC license |
Latest revision as of 16:38, 15 January 2022
Contents
Example 1
Evaluate the integral
Solution
Substitute . Then and we have
Applying the formula with we obtain
Example 2
Evaluate .
Solution
This integral requires two different methods to evaluate it. We get to those methods by splitting up the integral:
The first integral is handled straightforward; the second integral is handled by substitution, with . We handle each separately.
: Set , so and . We have
Combining these together, we have
Resources
- Integration into Inverse trigonometric functions using Substitution by The Organic Chemistry Tutor
- Integrating using Inverse Trigonometric Functions by patrickJMT
Licensing
Content obtained and/or adapted from:
- Integrals Resulting in Inverse Trigonometric Functions, LibreTexts: Mathematics under a CC BY-SA-NC license