Question

Solve :

${\int }^{}\frac{\left(3x-2\right)}{{\left(x+1\right)}^2\left(x+3\right)}dx$

Answer 1

$\begin{array}{c}{\int }^{}\frac{\left(3x-2\right)}{{\left(x+1\right)}^2\left(x+3\right)}dx\\ Let\frac{\left(3x-2\right)}{{\left(x+1\right)}^2\left(x+3\right)}=\frac{A}{\left(x+1\right)}+\frac{B}{{\left(x+1\right)}^2}+\frac{C}{\left(x+3\right)}\\ \Rightarrow \left(3x-2\right)\equiv A\left(x+1\right)\left(x+3\right)+B\left(x+3\right)+C{\left(x+1\right)}^2.......\left(i\right)\\ Puttingx=-13nobothsideof\left(i\right),wegetC=\frac{-11}{4}\\ Puttingx=-1onbothsidesof\left(i\right),wegetB=\frac{-5}{2}\\ A+C=0\Rightarrow A=-C=\frac{11}{4}\\ \therefore \frac{\left(3x-2\right)}{{\left(x+1\right)}^2\left(x+3\right)}=\frac{11}{4\left(x+1\right)}-\frac{5}{2{\left(x+1\right)}^2}-\frac{11}{4\left(x+3\right)}\\ \Rightarrow {\int }^{}\frac{\left(3x-2\right)}{{\left(x+1\right)}^2\left(x+3\right)}dx\\ =\frac{11}{4}{\int }^{}\frac{dx}{\left(x+1\right)}-\frac{5}{2}{\int }^{}\frac{1}{{\left(x+1\right)}^2}dx-\frac{11}{4}{\int }^{}\frac{dx}{\left(x+3\right)}\\ =\frac{11}{4}.\log \left|\frac{x+1}{x+3}\right|+\frac{5}{2\left(x+1\right)}+C\end{array}$