integration by parts trigonometric substitution
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Depending on the function we need to integrate, we substitute one of the following trigonometric expressions to simplify the integration:
For \displaystyle\sqrt{{{a}^{2}-{x}^{2}}}a2−x2, use \displaystyle{x}={a} \sin{\theta}x=asinθ
For \displaystyle\sqrt{{{a}^{2}+{x}^{2}}}a2+x2, use \displaystyle{x}={a} \tan{\theta}x=atanθ
For \displaystyle\sqrt{{{x}^{2}-{a}^{2}}}x2−a2, use \displaystyle{x}={a} \sec{\theta}x=asecθ
After we use these substitutions we'll get an integral that is "do-able".
For \displaystyle\sqrt{{{a}^{2}-{x}^{2}}}a2−x2, use \displaystyle{x}={a} \sin{\theta}x=asinθ
For \displaystyle\sqrt{{{a}^{2}+{x}^{2}}}a2+x2, use \displaystyle{x}={a} \tan{\theta}x=atanθ
For \displaystyle\sqrt{{{x}^{2}-{a}^{2}}}x2−a2, use \displaystyle{x}={a} \sec{\theta}x=asecθ
After we use these substitutions we'll get an integral that is "do-able".
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