Difference between revisions of "Integrals Resulting in Inverse Trigonometric Functions"
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<p>Evaluate the integral</p> | <p>Evaluate the integral</p> | ||
− | <p class="mt-align-center">\[ ∫\dfrac{dx}{\sqrt{4−9x^2}}.\nonumber\]</p> | + | <p class="mt-align-center"><math>\[ ∫\dfrac{dx}{\sqrt{4−9x^2}}.\nonumber\]</math></p> |
<p><strong>Solution</strong></p> | <p><strong>Solution</strong></p> | ||
− | <p>Substitute \( u=3x\). Then \( du=3\,dx\) and we have</p> | + | <p>Substitute <math>\( u=3x\)</math>. Then <math>\( du=3\,dx\)</math> 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 style="text-align: center;"><math>\[ ∫\dfrac{dx}{\sqrt{4−9x^2}}=\dfrac{1}{3}∫\dfrac{du}{\sqrt{4−u^2}}.\nonumber\]</math></p> |
− | <p>Applying the formula with \( a=2,\) we obtain</p> | + | <p>Applying the formula with <math>\( a=2,\)</math> 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> | + | <p class="mt-indent-3" style="text-align:center;"><math>\[ ∫\dfrac{dx}{\sqrt{4−9x^2}}=\dfrac{1}{3}∫\dfrac{du}{\sqrt{4−u^2}}=\dfrac{1}{3}\arcsin \left(\dfrac{u}{2}\right)+C=\dfrac{1}{3}\arcsin \left(\dfrac{3x}{2}\right)+C.\nonumber\]</math></p> |
==Resources== | ==Resources== |
Revision as of 13:55, 28 October 2021
Evaluate the integral
Failed to parse (syntax error): {\displaystyle \[ ∫\dfrac{dx}{\sqrt{4−9x^2}}.\nonumber\]}
Solution
Substitute Failed to parse (syntax error): {\displaystyle \( u=3x\)} . Then Failed to parse (syntax error): {\displaystyle \( du=3\,dx\)} and we have
Failed to parse (syntax error): {\displaystyle \[ ∫\dfrac{dx}{\sqrt{4−9x^2}}=\dfrac{1}{3}∫\dfrac{du}{\sqrt{4−u^2}}.\nonumber\]}
Applying the formula with 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 \( a=2,\)} we obtain
Failed to parse (syntax error): {\displaystyle \[ ∫\dfrac{dx}{\sqrt{4−9x^2}}=\dfrac{1}{3}∫\dfrac{du}{\sqrt{4−u^2}}=\dfrac{1}{3}\arcsin \left(\dfrac{u}{2}\right)+C=\dfrac{1}{3}\arcsin \left(\dfrac{3x}{2}\right)+C.\nonumber\]}
Resources
Integration into Inverse trigonometric functions using Substitution by The Organic Chemistry Tutor
Integrating using Inverse Trigonometric Functions by patrickJMT