Mathematics

$$\displaystyle \int \dfrac {1}{\sqrt {\sin^{3}x\sin(x+a)}}dx$$ is equal to


ANSWER

$$-2\cos ec\alpha \sqrt {\cos \alpha +\sin \alpha \cot x}+c$$


SOLUTION
$$\begin{array}{l} \int { \frac { { dx } }{ { \sqrt { { { \sin   }^{ 3 } }x\sin  \left( { x+a } \right)  }  } }  }  \\ =\int { \frac { { dx } }{ { \sqrt { { { \sin   }^{ 3 } }x\sin  x\left( { \cos  a+\cot  x\sin  a } \right)  }  } }  }  \\ =\int { \frac { { \cos  e{ c^{ 2 } }xdx } }{ { \sqrt { \cos  a+\sin  a\cot  x }  } }  }  \\ \cos  a+\sin  a\cot  =t \\ \frac { { dt } }{ { dx } } =-\sin  a\cos  e{ c^{ 2 } }x \\ -\frac { 1 }{ { \sin  a } } dt=\cos  e{ c^{ 2 } }xdx \\ =-\frac { 1 }{ { \sin  a } } \int { \frac { { dt } }{ { \sqrt { t }  } }  }  \\ =-\frac { 1 }{ { \sin  a } } \int { { t^{ \frac { { -1 } }{ 2 }  } }dt }  \\ =\frac { { -2 } }{ { \sin  a } } { t^{ \frac { 1 }{ 2 }  } }+C \\ =-2\cos  eca\sqrt { \cos  a+\sin  a\cot  x } +C \end{array}$$

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Single Correct Medium Published on 17th 09, 2020
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