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"><math>\[ &int;\dfrac{dx}{\sqrt{4&minus;9x^2}}.\nonumber\]</math></p>
+
<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>
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<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>\[ &int;\dfrac{dx}{\sqrt{4&minus;9x^2}}=\dfrac{1}{3}&int;\dfrac{du}{\sqrt{4&minus;u^2}}.\nonumber\]</math></p>
+
<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>\[ &int;\dfrac{dx}{\sqrt{4&minus;9x^2}}=\dfrac{1}{3}&int;\dfrac{du}{\sqrt{4&minus;u^2}}=\dfrac{1}{3}\arcsin \left(\dfrac{u}{2}\right)+C=\dfrac{1}{3}\arcsin \left(\dfrac{3x}{2}\right)+C.\nonumber\]</math></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>
  
 
==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