Question

Evaluate: $\int \frac{dx}{\left(5-8x-x^2\right)}dx$

Answer 1

$\begin{array}{c}Wehave\\ \int \frac{dx}{\left(5-8x-x^2\right)}dx=-\int \frac{dx}{\left(x^2+8x-5\right)}\\ =-\int \frac{dx}{\left\{\left(x^2+8x+16\right)-21\right\}}\\ =-\int \frac{dx}{\left\{{\left(x+4\right)}^2-{\left(\sqrt{21}\right)}^2\right\}}\\ =\int \frac{dx}{\left\{{\left(\sqrt{21}\right)}^2-{\left(x+4\right)}^2\right\}}\\ =\int \frac{dt}{\left\{{\left(\sqrt{21}\right)}^2-t^2\right\}}where\left(x+4\right)=t\\ =\frac{1}{2\sqrt{21}}.\log \left|\frac{\sqrt{21}+t}{\sqrt{21}-t}\right|+C\\ =\frac{1}{2\sqrt{21}}.\log \left|\frac{\sqrt{21}+4+x}{\sqrt{21}-4-x}\right|+C.\end{array}$

Hence, this is the required answer.