# Integrate the function$\frac{\large 2x}{\large 1+x^2}$

Toolbox:
• $(i)\;Method \;of \;substitution:$
• Given $\int f(x)dx$ can be transformed into another form by changing independent variable x to t by substituting x=g(t).
• Consider $I=\int f(x)dx.$
• Put x=g(t) so that $\frac{dx}{dt}=g'(t)$.
• dx=g'(t)dt.
• Thus $I=\int f(x)dx=\int f(g(t).g'(t))dt.$
$(ii)\;\int\frac{1}{x}dx=log x+c$.

Given $I=\int\frac{2x}{1+x^2}dx.$

Let $1+x^2=t.$

Differentiating on both sides we get,

2x dx=dt.

Substituing for $(1+x^2)$ and 2xdx we get,

$I=\int \frac{dt}{t}.$

Now integrating we get,

$I=log t+c$.

Now substituting back for t we get,

$log(1+x^2)+c$.

Hence $\int\frac{2x}{1+x^2}dx=log(1+x^2)+c.$