Question

$2\int \frac{tdt}{t^4+1}=\frac{1}{\sqrt{m}}ta{n}^{-1}{t}^2$.Find the value of $m$

Answer 1

$2\int \frac{tdt}{t^4+1}$

Let $t^2=M$

$\Rightarrow 2+dt=dM$ ( On differentiating w.r.t $t$ )

$\Rightarrow \frac{2}{2}\int \frac{dM}{M^2+1}={\tan }^{-1}M+C$

$={\tan }^{-1}\left(t^2\right)+C.$

Hence, the answer is ${\tan }^{-1}\left(t^2\right)+C.$