# Integration by trigonometric substitution

Jun 21st, 2015
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homesitemapmath interactivesmath blogblog sitemapaboutfeedbackTop of FormBottom of FormShare this page.Integration by 8Trigonometric SubstitutionIn this section, we see how to integrate expressions like(x2+9)3/2dxDepending on the function we need to integrate, we substitute one of the following trigonometric expressions to simplify the integration:For a2x2, use x=asinFor a2+x2, use x=atanFor x2a2, use x=asecAfter we use these substitutions we'll get an integral that is "do-able". Take note that we are not integrating trigonometric expressions (like we did earlier in Integration: The Basic Trigonometric Forms and Integrating Other Trigonometric Forms and Integrating Inverse Trigonometric Forms.Rather, on this page, we substitute a sine, tangent or secant expression in order to make an integral possible.Example 1(x2+9)3/2dxAnswer We can write the question as (32+x2)3/2dxIt's now in the form of the second substitution suggestion given above, that is:For a2+x2, use x=atan,with a=3.So we'll put x=3tan and this gives dx=3sec2dWe make the first substitution and simplify the denominator of the question before proceeding to integrate.We'll need to use

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