Question

How do you find the integral `int (x+1)/(x^2+2x+2)^3dx` using substitution?

Answer 1

The answer is `=-1/(4(x^2+2x+2))+C`

Explanation:

We use `intx^ndx=x^(n+1)/(n+1)+C (n!=0)`

Let `u=x^2+2x+2`

`du=(2x+2)dx`

`(x+1)dx=(du)/2`

`int((x+1)dx)/(x^2+2x+2)^3=1/2int(du)/(u^3)=1/2intu^(-3)du`

`=1/2*u^(-3+1)/(-3+1)`

`=1/2*u^(-2)/(-2)`

`=-1/(4u^2)`

`=-1/(4(x^2+2x+2))+C`

Answer 2

`-1/(4(x^2+2x+2)^2)+C`

Explanation:

You first decide which part of one looks like the other. If you make
`U= x^2+2x+2`
`dU= 2x+2`
Therefore, all you need to do is multiply the top by 2 and you have the same thing as your dU.
You then have multiply the inside by 2 but also the outside by 1/2.
Once you have this, you can then substitute your U and dU in your situation.
`1/2int_ . 1/((U)^3) du`

you final recieve after your done
`1/2(-1/(2u^2))`

Then you finally substitute your U back in the situation and you have

`-1/(4(x^2+2x+2)^2 )+ C`