Question

How do you integrate?

Answer 1

Use long division and the separation of terms.

Explanation:

Let

`I=int(5x^3-x^2+8)/(x^2+3x+12)dx`

Apply long division:

`I=int(5x-16-(12x-200)/(x^2+3x+12))dx`

Factor out the term where the numerator is a multiple of the derivative of the denominator:

`I=int(5x-16-(6(2x+3))/(x^2+3x+12)+218/(x^2+3x+12))dx`

Complete the square in the denominator:

`I=int(5x-16-(6(2x+3))/(x^2+3x+12)+872/((2x+3)^2+39))dx`

Integrate term by term:

`I=5/2x^2-16x-6ln|x^2+3x+12|+436/sqrt39tan^(-1)((2x+3)/sqrt39)+C`