Difference between revisions of "Integrals Resulting in Inverse Trigonometric Functions"
Line 7: | Line 7: | ||
<p>Evaluate the integral</p> | <p>Evaluate the integral</p> | ||
− | <p class="mt-align-center"><math>\[ | + | <p class="mt-align-center"><math>\[ \int\dfrac{dx}{\sqrt{4 - 9x^2}}.\]</math></p> |
<p><strong>Solution</strong></p> | <p><strong>Solution</strong></p> | ||
Line 13: | Line 13: | ||
<p>Substitute <math>\( u=3x\)</math>. Then <math>\( du=3\,dx\)</math> 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;"><math>\[ | + | <p style="text-align: center;"><math>\[ \int;\dfrac{dx}{\sqrt{4 - 9x^2}}=\dfrac{1}{3}\int\dfrac{du}{\sqrt{4 - u^2}}.\nonumber\]</math></p> |
<p>Applying the formula with <math>\( a=2,\)</math> we obtain</p> | <p>Applying the formula with <math>\( a=2,\)</math> we obtain</p> | ||
− | <p class="mt-indent-3" style="text-align:center;"><math>\[ | + | <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> |
==Resources== | ==Resources== |
Revision as of 14:17, 28 October 2021
Evaluate the integral
Failed to parse (syntax error): {\displaystyle \[ \int\dfrac{dx}{\sqrt{4 - 9x^2}}.\]}
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 \[ \int;\dfrac{dx}{\sqrt{4 - 9x^2}}=\dfrac{1}{3}\int\dfrac{du}{\sqrt{4 - u^2}}.\nonumber\]}
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 (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 \[ \int;\dfrac{dx}{\sqrt{4 - 9x^2}}=\dfrac{1}{3}\int\dfrac{du}{\sqrt{4 - u^2}}=\dfrac{1}{3}\arcsin \left(\dfrac{u}{2}\right)+C=\dfrac{1}{3}\arcsin \left(\dfrac{3x}{2}\right)+C.\]}
Resources
Integration into Inverse trigonometric functions using Substitution by The Organic Chemistry Tutor
Integrating using Inverse Trigonometric Functions by patrickJMT