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Q2  Integrate the functions \frac{1}{\sqrt { 1+ 4 x^2 }}

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\frac{1}{\sqrt { 1+ 4 x^2 }}
let suppose 2x = t 
therefore 2dx = dt

\int \frac{1}{\sqrt{1+4x^2}} =\frac{1}{2}\int \frac{dt}{1+t^2}
                              \\=\frac{1}{2}[\log\left | t+\sqrt{1+t^2} \right |]+C\\ =\frac{1}{2}\log\left | 2x+\sqrt{4x^2+1} \right |+C.................using formula \int\frac{1}{\sqrt{x^2+a^2}}dt = \log\left | x+\sqrt{x^2+a^2} \right |

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manish

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