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Title: Im stuck at integrating algebraic substitution Post by: Chill on Feb 5, 2019 Can you solve this pls. By using algebraic substitution.
Integral of x^2/(x^2+1)^3/2 Title: Re: Im stuck at integrating algebraic substitution Post by: bio_man on Feb 5, 2019 Our function is:
\(\int \frac{x^2}{\left(x^2+1\right)^{\frac{3}{2}}}dx\) I looked at this for 20 minutes, and realized that algebraic substitution is NOT the right way to do it. Instead, you have to perform trigonometric substitution: \(x=\tan \left(u\right)\) Solve for \(u\), then find derivative with respect to x: \(\tan ^{-1}x=u\) \(\frac{du}{dx}=\sec ^2\left(u\right)\) Solve for dx: \(du=\sec ^2\left(u\right)\cdot dx\) Now your expression is: \(\int \frac{\tan ^2\left(u\right)}{\left(\tan ^2u+1\right)^{\frac{3}{2}}}\sec ^2\left(u\right)du\) At the bottom: Simplify using \(\tan ^2(u)+1=\sec ^2(u)\) We get: \(\int \frac{\tan ^2\left(u\right)}{\left[\sec ^2\left(u\right)\right]^{\frac{3}{2}}}\sec ^2\left(u\right)du\) \(\int \frac{\tan ^2\left(u\right)}{\sec ^3\left(u\right)}\sec ^2\left(u\right)du\) \(\int \frac{\tan ^2\left(u\right)}{\sec \left(u\right)}du\) Same as: \(\int \cos \left(u\right)\tan ^2\left(u\right)du\) . . . Please let me know if you're following |