Question

How do you find the antiderivative of `int 1/(x^2+10x+25) dx`?

Answer 1

`-1/(x+5)+C`

Explanation:

Notice the denominator can be factored:

`int1/(x^2+10x+25)dx=int1/(x+5)^2dx=int(x+5)^-2dx`

Now we can use substitution. Let `u=x+5`. Thus, `du=dx`. Substituting them in:

`=intu^-2du`

Now use the power rule for integration `intu^ndu=u^(n+1)/(n+1)+C`

`=u^(-2+1)/(-2+1)+C=u^(-1)/(-1)+C=-1/u+C=-1/(x+5)+C`