Evaluate integration using rule

Question

Please see question

https://i.stack.imgur.com/Hx4CT.jpg

I can divide the first part ok, but how do i get the answer for the 2nd part using the substitution

Answer 1

Hint: Your integrand is equal to $${x}^{3}-x+{\frac {x+1}{{x}^{2}+1}}$$

Answer 2

$\int \frac {x^{5}+1} {x^{2}+1}=\int \frac {x^{5}} {x^{2}+1}+\int \frac 1 {x^{2}+1}$. In the first term put $t=x^{2}+1$ to get $\frac 1 2 \int \frac {(t-1)^{2}} t dt$ and you can evaluate this by expanding $(t-1)^{2}$. The second term is $\arctan x$.