Integrate the function 1 1 + 4 x^2 - Vidyadip

Question

Integrate the function $\frac{1}{{\sqrt {1 + 4{x^2}} }}$

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Answer

$\int {\frac{1}{{\sqrt {1 + 4{x^2}} }}} dx$ $= \int {\frac{1}{{\sqrt {{{\left( {2x} \right)}^2} + {{\left( 1 \right)}^2}} }}} dx$
$= \frac{{\log \left| {\left( {2x} \right) + \sqrt {{{\left( {2x} \right)}^2} + {1^2}} } \right|}}{2}+c$ ...[dividing by 2 as coefficient of x is 2]
$\left[ {\because \int {\frac{1}{{\sqrt {{x^2} + {a^2}} }}dx = \log \left| {x + \sqrt {{x^2} + {a^2}} } \right|} } \right]$
$= \frac{1}{2}\log \left| {2x + \sqrt {4{x^2} + 1} } \right| + c$

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