Question

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

Answer 1

$I=\int \frac{dx}{1+2(x+2{)}^2}$ put x+2=t dx=dt Differentiating,

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

$=\frac{1}{2}\int \frac{dt}{\frac{1}{2}+t^2}=\frac{1}{2}\int \frac{dt}{\left(\frac{1}{\sqrt{2}}\right)+y^2}$

$=\frac{1}{2}\times \frac{1}{\left(\frac{1}{\sqrt{2}}\right)}ta{n}^{-1}\left(\frac{t}{\frac{1}{\sqrt{2}}}\right)+c$ $(\because \int \frac{dx}{a^2+x^2}=\frac{1}{a}ta{n}^{-1}(\frac{x}{a})+c)$

$=\frac{1}{\sqrt{2}}ta{n}^{-1}\left(\sqrt{2t}\right)+c$