Question

Integrate `int(5x+7)/(x^2+4x+8)dx`?

Answer 1

`int(5x+7)/(x^2+4x+8)dx=5/2ln|x^2+4x+8|-24tan^(-1)(x/2+1)`

Explanation:

`int(5x+7)/(x^2+4x+8)dx` can be written as

`int5/2*(2x+4)/(x^2+4x+8)dx-int3/(x^2+4x+8)dx`

= `5/2int(2x+4)/(x^2+4x+8)dx-3int1/(x^2+4x+8)dx`

= `5/2ln|x^2+4x+8|-3int1/((x+2)^2+4)dx`

• (as `d/dx(x^2+4x+8)=2x+4`)

= `5/2ln|x^2+4x+8|-12int1/(((x+2)/2)^2+1)dx`

Now let `(x+2)/2=tantheta` or `1/2dx=sec^2thetad theta` and then

`int1/(((x+2)/2)^2+1)dx=int1/sec^2thetaxx2sec^2thetad theta`

= `int2d theta=2theta=2tan^(-1)(x/2+1)`

Hence `int(5x+7)/(x^2+4x+8)dx`

= `5/2ln|x^2+4x+8|-24tan^(-1)(x/2+1)`