Indefinite Integrals — Maths STD 12 Science — Question

$\Rightarrow\text{x}=\text{A}(\text{x}^2+1)+(\text{Bx}+\text{C})(\text{x}+1)$

Equation similar terms, we get

$\text{A}+\text{B}=0,\text{B}+\text{C}=1,\text{A}+\text{C}=0$

Solving, we get, $\text{A}=-\frac{1}{2},\text{B}=\frac{1}{2},\text{C}=\frac{1}{2}$

Thus,

$\text{I}=-\frac{1}{2}\int\frac{\text{dx}}{\text{x}+1}+\frac{1}{2}\int\frac{\text{x dx}}{\text{x}^2+1}+\frac{1}{2}\int\frac{\text{dx}}{\text{x}^2+1}$

$\text{I}=-\frac{1}{2}\log|\text{x}+1|+\frac{1}{4}\log|\text{x}^2+1|+\frac{1}{2}\tan^{-1}\text{x}+\text{c}$