Question

Solve $\int \frac{1}{3x^2+13x+10}dx$

Answer 1

$\int \frac{1}{3x^2+13x+10}dx$

$=\int \frac{dx}{3x^2+3x+10x+10}$

$=\int \frac{dx}{(3x+10)(x+1)}$

$=\int \frac{A}{3x+10}+\frac{13}{x+1}dx=\frac{1}{7}\int \frac{-3dx}{3x+10}+\frac{dx}{x+1}$

$=\frac{1}{7}[ln\left(\frac{1}{3x+10}\right)+ln(x+1)]+c$

$=\frac{1}{7}ln\frac{(x+1)}{(3x+10)}+C$

we know $Ax+A+3Bx+10B=1$

$\Rightarrow A+10B=1$

$A+3B=0$

$\Rightarrow 7B=1$

$\Rightarrow B=\frac{1}{7}|A=\frac{-3}{7}$