Can You Solve This Integral Without Using Integration by Parts?

[raincheck]

"Evaluate the integral [0,1] x^3/sqrt[x^2 + 1] by integration by parts"

I know I have to use the integration by parts equation, but I don't know what to make u and what to make dv..

 

[Hootenanny]

Try rewriting the integral thus;

$$\int^{1}_{0}\; \frac{x^{3}}{\sqrt{x^2+1}} \; dx = \int^{1}_{0}\; x^3 (x^2 +1)^{-\frac{1}{2}} \;dx$$

Now, to determine which term to make u and dv, think about which one will simplify your expression most when you differentiated it and set this to u.

 

[Repetit]

You could start by rewriting the integrand:

$$\frac{x^3}{\sqrt{x^2+1}}=x^3 (x^2+1)^{-1/2} = [x^{-6} (x^2+1)]^{-1/2}$$

And the apply the integration by parts formula. That should make it easier.

 

[raincheck]

OH okay, thank you!

 

[Max Eilerson]

Do you have to do it by parts? Much easier to substitute $$u = x^2$$