Find the integral of the function 1√4x2+64x+100 ln|x+8

SOLUTION

$\int \frac{dx}{\sqrt{4x^2+64x+100}}$
We will try to express this as a perfect square and proceed
$\int \frac{dx}{\sqrt{4x^2+64x+100}}=\int \frac{dx}{\sqrt{4(x^2+16x+25)}}$
=$\int \frac{dx}{2\sqrt{(x+4{)}^2+9)}}$
This integral is of the form $\int \frac{dx}{\sqrt{x^2+a^2}}$ which is equal to $ln|x+\sqrt{x^2+a^2}|+c$
$\Rightarrow \int \frac{dx}{2\sqrt{(x+4{)}^2+3^2)}}=ln|(x+4)+\sqrt{(x+4{)}^2+3^2}|+c$