Question

Integrate `(3x^2+cosx)/(x^3+sinx)`?

Answer 1

`int \ (3x^2+cosx)/(x^3+sin x) \ dx = ln|x^3+sin x| + C`

Explanation:

We want to find:

`I = int \ (3x^2+cosx)/(x^3+sin x) \ dx`

We can integral of the expression using a simple substitution: Let

`u = x^3+sin x => (du)/dx = 3x^2 + cosx`

Substituting into the integral we get:

`I = int \ 1/u \ du`

Which we can now integrate to get:

`I = ln|u| + C`

And restoring the substitution we get:

`I = ln|x^3+sin x| + C`