Question about Integral (Calculus)!?

Question

What is the Integral of 1/((x+5)^2(x-1)) AND (x^3)/(x^2+1) ?

Show Detail and Steps Please!

Thank to all !

Answer 1

The key to solving this is to use u substitution.
∫(x^3)/(x^2+1) dx

Use u substitution for x^2 + 1:
u = x^2 + 1
du = 2x dx
1/2 du = x dx

Since we got 1/2 du = x dx, we need to separate out x dx to let us substitute:
∫x^2/(x^2 + 1) * x dx
∫x^2/u * 1/2 du
1/2 ∫x^2/u du

Now we need the x^2 converted in terms of u.
u = x^2 + 1
u - 1 = x^2

Plug that in:
1/2 ∫(u - 1)/u * du

Distribute out the parenthesis:
1/2 ∫(u/u) - (1/u) du
1/2 ∫1 - 1/u du

Integrate:
1/2 [u - ln u] + C
1/2 ((x^2 + 1) - ln (x^2 + 1)) + C (solution!)