Question

Solve:
$\int \frac{5x}{(x+1)(x^2+2)}dx$

Answer 1

$I=\int \frac{5x}{(x+1)(x^2+2)}=\int \frac{A}{x+1}+\frac{B+Cx}{x^2+2}dx$

$\Rightarrow A{x}^2+2A+Bx+B+C{x}^2+Cx=5x$

$\Rightarrow A+C=0$_____(1)

$2A+B=0$____(2)

$B+C=5$_____(3)

$\Rightarrow A=5,B=10,C=-5$

$\Rightarrow I\int \frac{5}{x+1}dx+\int \frac{10}{x^2+1}dx-\int \frac{5x}{x^2+1}dx$

$\Rightarrow I=5ln|x+1|+10ta{n}^{-1}x-\frac{5}{2}ln|x^2+1|+K$

$K=$ constant