Mathematics

Integrate:
$$\int e^x\sin  x.dx$$


SOLUTION
$$I=\int { { e }^{ x }\sin xdx } \\ ={ e }^{ x }\int { \sin xdx } +\int { \left[ \dfrac { d }{ dx } { e }^{ x }.\int { \sin xdx }  \right] dx } \\ =-{ e }^{ x }\cos x-\int { { e }^{ x }\cos xdx } \\ =-{ e }^{ x }\cos x-\left[ { e }^{ x }\int { \cos xdx } +\int { \left[ \dfrac { d }{ dx } { e }^{ x }.\int { \cos xdx }  \right] dx }  \right] \\ =-{ e }^{ x }\cos x-{ e }^{ x }\sin x-\int { { e }^{ x }\sin xdx } \\ I=-{ e }^{ x }\cos x-{ e }^{ x }\sin x-I\\ 2I=-{ e }^{ x }(\cos x+\sin x)\\ I=-\dfrac { { e }^{ x } }{ 2 } (\cos x+\sin x)$$
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Subjective Medium Published on 17th 09, 2020
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