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>
  
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<p>Evaluate the integral</p>
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<p class="mt-align-center">\[ &int;\dfrac{dx}{\sqrt{4&minus;9x^2}}.\nonumber\]</p>
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<p><strong>Solution</strong></p>
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<p>Substitute \( u=3x\). Then \( du=3\,dx\) and we have</p>
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<p style="text-align: center;">\[ &int;\dfrac{dx}{\sqrt{4&minus;9x^2}}=\dfrac{1}{3}&int;\dfrac{du}{\sqrt{4&minus;u^2}}.\nonumber\]</p>
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<p>Applying the formula with \( a=2,\) we obtain</p>
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<p class="mt-indent-3" style="text-align:center;">\[ &int;\dfrac{dx}{\sqrt{4&minus;9x^2}}=\dfrac{1}{3}&int;\dfrac{du}{\sqrt{4&minus;u^2}}=\dfrac{1}{3}\arcsin \left(\dfrac{u}{2}\right)+C=\dfrac{1}{3}\arcsin \left(\dfrac{3x}{2}\right)+C.\nonumber\]</p>
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==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