How do you integrate int x^3sqrt(1+x^2) using integration by parts?

1 Answer
mason m
Nov 8, 2016

((1+x^2)^(3/2)(3x^2-2))/15+C

Explanation:

I=intx^3sqrt(1+x^2)dx

Before trying any integration by parts, we should attempt more basic substitutions to simplify the integral. Let u=1+x^2. This implies that du=2xdx and x^2=u-1.

Rearranging the integral:

I=1/2intx^2sqrt(1+x^2)(2xdx)=1/2int(u-1)sqrtudu

So, we see that integration by parts won't be necessary at all!

I=1/2int(u^(3/2)-u^(1/2))du=1/2(u^(5/2)/(5/2)-u^(3/2)/(3/2))

color(white)I=1/2(2/5u^(5/2)-2/3u^(3/2))=u^(5/2)/5-u^(3/2)/3=(3u^(5/2)-5u^(3/2))/15

color(white)I=(u^(3/2)(3u-5))/15=((1+x^2)^(3/2)(3x^2-2))/15+C

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