Question

Integrate : $\frac{dy}{dx}=\frac{1}{2(1+x^2)}$

Answer 1

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

$\Rightarrow 2dy=\frac{dx}{1+x^2}$

$\Rightarrow 2\int dy=\int \frac{dx}{1+x^2}$

Integrating both sides

$\Rightarrow 2y={\tan }^{-1}x+c$

since $\int \frac{dx}{1+x^2}={\tan }^{-1}x+c$

$\therefore y=\frac{1}{2}{\tan }^{-1}x+\frac{c}{2}$

or $y=\frac{1}{2}{\tan }^{-1}x+c_1$ where $c_1=\frac{c}{2}$