Question

Evaluate the integration of : (co3x-x^2)(sin3x-x^3)dx?

Answer 1

`I=1/3ln(abs(sin(3x)-x^3))+C`

Explanation:

We want to solve

`I=int(cos(3x)-x^2)/(sin(3x)-x^3)dx`

Make a substitution

`color(blue)(u=sin(3x)-x^3=>du=3(cos(3x)-x^2)dx`

`I=int(cos(3x)-x^2)/(u*3(cos(3x)-x^2))du`

`color(white)(I)=1/3int1/udu`

`color(white)(I)=1/3ln(abs(u))+C`

Substitute back `color(blue)(u=sin(3x)-x^3)`

`I=1/3ln(abs(sin(3x)-x^3))+C`