Mathematics

Solve :
$$\int x^{ \sin x} \left( \dfrac {sinx}{x} + \cos x . logx \right) dx $$


SOLUTION
$$\begin{array}{l} \int { { x^{ \sin  x } }\left( { \frac { { \sin  x } }{ x } +\cos  x\log  x } \right)  } dx \\ y={ x^{ \sin  x } } \\ \log  y=\sin  x\log  x \\ \frac { 1 }{ y } \frac { { dy } }{ { dx } } =\cos  x\log  x+\frac { { \sin  x } }{ x }  \\ \frac { { dy } }{ { dx } } ={ x^{ \sin  x } }\left[ { \cos  x{ { logx } }+\frac { { \sin  x } }{ x }  } \right]  \\ \int { dy } =y+C \\ ={ x^{ \sin  x } }+C \end{array}$$
Hence,
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Subjective Medium Published on 17th 09, 2020
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