Question

Evaluate:

${\int }_0^1\frac{2x}{x^2+1}dx$

Answer 1

${\int }_0^1\frac{2x}{x^2+1}dx$

Let $t=x^2+1⟹dt=2xdx$

$⟹x\to 0\to 1$ so

$t\to 1\to 2$

${\int }_1^2\frac{1}{t}dt$

By using $\int \frac{1}{n}=\log t+c$

$={\log t|}_1^2$

$\Rightarrow \log 2-\log 1=\log 2$