Indefinite Integrals — Maths STD 12 Science — Question

$=\int3\tan^{-1}(\text{x})\text{dx}$

$=3\int\big[\tan^{-1}(\text{x})\times1\big]\text{dx}$

$=3\Big[\tan^{-1}\text{x}\times\text{x}-\int\frac{1}{1+\text{x}^2}\times\text{x dx}\Big]$

$=3\text{x}\tan^{-1}\text{x}-3\int\frac{\text{x}}{1+\text{x}^2}\text{dx}$

Let $1+\text{x}^2=\text{t}$

$\Rightarrow2\text{x dx = dt}$

Then,

$\text{I}=3\text{x}\tan^{-1}\text{x}-\frac{3}{2}\int\frac{\text{dt}}{\text{t}}$

$=3\text{x}\tan^{-1}\text{x}-\frac{3}{2}\log|\text{t}|+\text{C}$

$=3\text{x}\tan^{-1}\text{x}-\frac{3}{2}\log|1+\text{x}^2|+\text{C}$