Question

Evaluate:

$\underset{0}{\overset{1}{\int }}\frac{x}{x^2+1}dx$

Answer 1

$\underset{0}{\overset{1}{\int }}\frac{x}{x^2+1}dx$

Let $x^2+1=t$

$\Rightarrow 2xdx=dt$

When $x=0,t=1$ and when $x=1,t=2$.

Therefore,

$\underset{0}{\overset{1}{\int }}\frac{x}{x^2+1}dx=\frac{1}{2}\underset{1}{\overset{2}{\int }}\frac{dt}{t}$

$=\frac{1}{2}[\log 2-\log 1]=\frac{1}{2}\log 2$