Difference between revisions of "Integrals Resulting in Inverse Trigonometric Functions"
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<math> \int\frac{du}{u\sqrt{u^2 - a^2}} =\frac{1}{a}\arcsec \left(\dfrac{|u|}{a}\right) + C </math> | <math> \int\frac{du}{u\sqrt{u^2 - a^2}} =\frac{1}{a}\arcsec \left(\dfrac{|u|}{a}\right) + C </math> | ||
+ | <p>Evaluate the integral</p> | ||
+ | |||
+ | <p class="mt-align-center">\[ ∫\dfrac{dx}{\sqrt{4−9x^2}}.\nonumber\]</p> | ||
+ | |||
+ | <p><strong>Solution</strong></p> | ||
+ | |||
+ | <p>Substitute \( u=3x\). Then \( du=3\,dx\) and we have</p> | ||
+ | |||
+ | <p style="text-align: center;">\[ ∫\dfrac{dx}{\sqrt{4−9x^2}}=\dfrac{1}{3}∫\dfrac{du}{\sqrt{4−u^2}}.\nonumber\]</p> | ||
+ | |||
+ | <p>Applying the formula with \( a=2,\) we obtain</p> | ||
+ | |||
+ | <p class="mt-indent-3" style="text-align:center;">\[ ∫\dfrac{dx}{\sqrt{4−9x^2}}=\dfrac{1}{3}∫\dfrac{du}{\sqrt{4−u^2}}=\dfrac{1}{3}\arcsin \left(\dfrac{u}{2}\right)+C=\dfrac{1}{3}\arcsin \left(\dfrac{3x}{2}\right)+C.\nonumber\]</p> | ||
+ | |||
+ | ==Resources== | ||
[https://youtu.be/AE-0gXXx_j0 Integration into Inverse trigonometric functions using Substitution] by The Organic Chemistry Tutor | [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 | [https://youtu.be/MdsAvt9y5ds Integrating using Inverse Trigonometric Functions] by patrickJMT |
Revision as of 13:53, 28 October 2021
Evaluate the integral
\[ ∫\dfrac{dx}{\sqrt{4−9x^2}}.\nonumber\]
Solution
Substitute \( u=3x\). Then \( du=3\,dx\) and we have
\[ ∫\dfrac{dx}{\sqrt{4−9x^2}}=\dfrac{1}{3}∫\dfrac{du}{\sqrt{4−u^2}}.\nonumber\]
Applying the formula with \( a=2,\) we obtain
\[ ∫\dfrac{dx}{\sqrt{4−9x^2}}=\dfrac{1}{3}∫\dfrac{du}{\sqrt{4−u^2}}=\dfrac{1}{3}\arcsin \left(\dfrac{u}{2}\right)+C=\dfrac{1}{3}\arcsin \left(\dfrac{3x}{2}\right)+C.\nonumber\]
Resources
Integration into Inverse trigonometric functions using Substitution by The Organic Chemistry Tutor
Integrating using Inverse Trigonometric Functions by patrickJMT