Question

Evaluate: $\int \frac{1}{1+(2x+1{)}^2}dx,x\in R$

Answer 1

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

$\Rightarrow 2x+1=t$

$dx=\frac{dt}{2}$

$\frac{1}{2}\int \frac{dt}{1+t^2}$

$\Rightarrow \frac{1}{2}{\tan }^{-1}\left(t\right)+c$

$=\frac{1}{2}{\tan }^{-1}(2x+1)+c$