Question

Evaluate :
$\int \frac{x+1}{x^2+3x+12}dx$

Answer 1

$\int \frac{x+1}{x^2+3x+12}dx$
$x+1=A\frac{d}{dx}\left(x^2+3x+12\right)+B$
$x+1=2Ax+3Ax+B$
$1=2A1=3A+B$
$A=\frac{1}{2}1=3\times \frac{1}{2}+BB=-\frac{1}{2}$
$\int \frac{x+1}{x^2+3x+12}dx=\frac{1}{2}\int \frac{\left(2x+3\right)}{x^2+3x+2}dx-\frac{1}{2}\int \frac{1}{x^2+3x+2}dx$
$=\frac{1}{2}\log \left[x^2+3x+2\right]-\frac{1}{2}\frac{dx}{x^2+3x+\frac{9}{4}-\frac{9}{4}+2}$
$=\frac{1}{2}\log \left[x^2+3x+2\right]-\frac{1}{2}\frac{dx}{{\left(x+\frac{3}{4}\right)}^2-{\left(\frac{1}{2}\right)}^2}$