Question

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

Answer 1

We have,

$I=\int \frac{3x+2}{(x+1{)}^2(x+2)}dx$

Let

$\frac{3x+2}{(x+1{)}^2(x+2)}=\frac{A}{x+1}+\frac{B}{(x+1{)}^2}+\frac{C}{x+2}$

$3x+2=A(x+1)(x+2)+B(x+2)+C(x+1{)}^2$

$3x+2=A(x^2+3x+2)+B(x+2)+C(x^2+2x+1)$ $........\left(1\right)$

On solving equation $\left(1\right)$, we get

$A=4,B=-1,C=-4$

Therefore,

$I=\int \frac{4}{x+1}dx-\int \frac{1}{(x+1{)}^2}dx-\int \frac{4}{x+2}dx$

$I=4\ln (x+1)+\frac{1}{x+1}-4\ln (x+2)+C$

Hence, this is the answer.