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Im stuck at integrating algebraic substitution
Chill
Chill
wrote...
2 weeks ago
2 weeks ago
Im stuck at integrating algebraic substitution
Can you solve this pls. By using algebraic substitution.
Integral of x^2/(x^2+1)^3/2
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bio_man
wrote...
#1
Educator
2 weeks ago
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:
\(\int \cos \left(u\right)\tan ^2\left(u\right)du\)
.
.
.
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