#### explain solution RD Sharma class 12 Chapter 18 Indefinite Integrals exercise 18.30 question 56

$\log \left|x^{2}+1\right|-\log \left|x^{2}+2\right|+\frac{1}{\left(x^{2}+2\right)}+C$

Hint:

To solve this integration, we use partial fraction method

Given:

$\int \frac{2 x}{\left(x^{2}+1\right)\left(x^{2}+2\right)^{2}} d x$

Explanation:

Let

$I=\int \frac{2 x}{\left(x^{2}+1\right)\left(x^{2}+2\right)^{2}} d x$

Let

\begin{aligned} &x^{2}=y \\ &2 x d x=d y \\ &I=\int \frac{d y}{(y+1)(y+2)^{2}} \\ &\frac{1}{(y+1)(y+2)^{2}}=\frac{A}{y+1}+\frac{B}{y+2}+\frac{C}{(y+2)^{2}} \\ &1=A(y+2)^{2}+B(y+2)(y+1)+C(y+1) \end{aligned}

Put $y=-2$

\begin{aligned} &1=A+(0)+(0) \\ &A=1 \end{aligned}

Put $y=0$

\begin{aligned} &1=4 A+2 B+C \\ &1=4+2 B-1 \\ &1=2 B+3 \\ &-2=2 B \\ &B=-1 \end{aligned}

\begin{aligned} &\frac{1}{(y+1)(y+2)^{2}}=\frac{1}{y+1}-\frac{1}{(y+2)}-\frac{1}{(y+2)^{2}} \\ &I=\int \frac{d y}{1+y}-\int \frac{d y}{y+2}-\int \frac{d y}{(y+2)^{2}} \\ &I=\log |1+y|-\log |y+2|+\frac{1}{(y+2)}+C \end{aligned}

As $y=x^{2}$

$I=\log \left|x^{2}+1\right|-\log \left|x^{2}+2\right|+\frac{1}{\left(x^{2}+2\right)}+C$