Question

Integrate
$\int \frac{dx}{\left(x+1\right)\left(x+5\right)}$

Answer 1

$\int \frac{dx}{(x+1)(x+5)}Let,\frac{1}{(x+1)(x+5)}=\frac{A}{x+1}+\frac{B}{x+5}....\left(1\right)\frac{1}{(x+1)(x+5)}=\frac{(x+5)A+(x+1)B}{(x+1)(x+5)}1=(x+5)A+(x+1)B1=Ax+5A+Bx+B1=(A+B)x+(5A+B)EquatingcoefficentsofxbothsideswegetA+B=\mathrm{0....}\left(2\right)and5A+B=\mathrm{1.....}\left(3\right)solvingaboveequation\left(2\right)and\left(3\right)wegetA=\frac{1}{4}andB=-\frac{1}{4}PutvaluesofAandBin\left(1\right)weget\frac{1}{(x+1)(x+5)}=\frac{1}{4(x+1)}-\frac{1}{4(x+5)}Now,\int \frac{dx}{(x+1)(x+5)}=\int \frac{dx}{4(x+1)}-\int \frac{dx}{4(x+5)}=\frac{1}{4}\left[\ln (x+1)-\ln (x+5)\right]=\frac{1}{4}\ln \left(\frac{x+1}{x+5}\right)$