Mathematics

Solve:$$\displaystyle \int\dfrac{1}{\sqrt{1+4x^2}}dx$$


SOLUTION
$$\int { \dfrac { 1 }{ \sqrt { 1+4{ { x }^{ 2 } } }  } dx } \\ =\int { \dfrac { 1 }{ \sqrt { 4\left( \dfrac { 1+4{ x }^{ 2 } }{ 4 }  \right)  }  } dx } \\ =\int { \left[ \dfrac { 1 }{ 2 } .\dfrac { 1 }{ \sqrt { \dfrac { 1 }{ 4 } +{ x }^{ 2 } }  }  \right] dx } \\ =\dfrac { 1 }{ 2 } \int { \dfrac { 1 }{ \sqrt { { \left( \dfrac { 1 }{ 2 }  \right)  }^{ 2 }+{ x }^{ 2 } }  } dx } \\ =\dfrac { 1 }{ 2 } \left[ \dfrac { 1 }{ \dfrac { 1 }{ 2 }  } { \tan }^{ -1 }\left( \dfrac { x }{ \dfrac { 1 }{ 2 }  }  \right)  \right] +C\\ ={ \tan }^{ -1 }2x+C$$
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Subjective Medium Published on 17th 09, 2020
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