Question

How do you integrate?

Answer 1

Use the substitution `x+3=3sectheta`.

Explanation:

Let

`I=int(x-3)/sqrt(x^2+6x)dx`

Complete the square in the square root:

`I=int(x-3)/sqrt((x+3)^2-9)dx`

Apply the substitution `x+3=3sectheta`:

`I=int(3sectheta-6)/(3tantheta)(3secthetatanthetad theta)`

Simplify:

`I=3int(sec^2theta-2sectheta)d theta`

Integrate directly:

`I=3(tantheta-2ln|sectheta+tantheta|)+C`

Rearrange:

`I=3tantheta-6ln|3sectheta+3tantheta|+C`

Reverse the substitution:

`I=sqrt(x^2+6x)-6ln|x+3+sqrt(x^2+6x)|+C`