How to Solve Tricky Integrals Involving x^3*sin(x^2) and Rational Functions?

[Lodve]

Hi I'm struggling solving this integration:
$$x^3 * sin x^2$$
My idea to solve this one is to apply integration by parts, but I can't get through using it.

$$\frac{4x+2}{x^2(x^2+2)}$$ this seems a hard nut to crack.

 

[kbaumen]

If you want to use integration by parts for the first one, there is a certain substitution that would make it quite simple. Can you see it?

 

[Lodve]

Hmmm not sure, what if a let the u equal to x^2 and derive it so that I get 2x?

 

[Mark44]

For integration by parts, I would start by breaking the product up into x² and xsin(x²).

 

[JonF]

u=x^2
du=dx*2x

so your argument would be:

1/2u*sin(u)du

which is easier to do parts on.

 

[gomunkul51]

For the second integral, use Partial Fraction Decomposition:

$$\frac{4x+2}{x^2(x^2+2)} = \frac{Ax+B}{x^2}+\frac{Cx+D}{(x^2+2)}$$

Each of the resulting fraction could be integrated by u, du substitution.

http://en.wikipedia.org/wiki/Partial_fraction