Definite integral involving rational function and 𝜋

[ahilan]

Mod note: Moved from technical math forum section. New member warned that homework-type posts must be posted in one of the homework sections and that some work must be shown.

I have tried to solve this integration but i can't, can somebody figure out a way to do this..

 

[martinbn]

$u=x^2+\pi +1$

 

[MidgetDwarf]

Typically , for these types of problem. A good technqie to think about is u-substitution and let u be the more complicated function.

 

[Gavran]

I have tried to solve this integration but i can't, can somebody figure out a way to do this..

Two techniques should be applied in this case. The first one is the substitution as proposed in post #2, and the second one is the integration by partial fractions. The order in which techniques should be used first is irrelevant. If the integration by partial fractions is used first, there will be the next equation. $$ \int_{0}^{\sqrt{\pi-1}}\frac{16\pi^2x^3}{(x^2+\pi+1)^3}dx=\int_{0}^{\sqrt{\pi-1}}\frac{16\pi^2x}{(x^2+\pi+1)^2}dx-\int_{0}^{\sqrt{\pi-1}}\frac{16\pi^3x}{(x^2+\pi+1)^3}dx-\int_{0}^{\sqrt{\pi-1}}\frac{16\pi^2x}{(x^2+\pi+1)^3}dx $$

 

[PeroK]

Two techniques should be applied in this case. The first one is the substitution as proposed in post #2, and the second one is the integration by partial fractions.

After the proposed u-substitution, the partial fractions step is trivial. In any case, it must be much simpler to do the u-substitution first.