What is the integral of e^x(sin(x))^2 ?

The integral of e^x(sin(x))^2 is e^xsin^2(x)+2/5 e^xcos(2x)-1/5 e^xsin(2x)+C

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integral of e^x(sin(x))^2

\int e^{x} (sin (x))^{2} d x

e^{x} sin^{2} (x) + \frac{2}{5} e^{x} cos (2 x) - \frac{1}{5} e^{x} sin (2 x) + C

Solution steps

\int e^{x} (sin (x))^{2} d x

Simplify

= \int e^{x} sin^{2} (x) d x

Apply Integration By Parts

: sin^{2} (x) e^{x} - \int sin (2 x) e^{x} d x

= sin^{2} (x) e^{x} - \int sin (2 x) e^{x} d x

\int sin (2 x) e^{x} d x = - \frac{2 e ^{x} cos ( 2 x )}{5} + \frac{e ^{x} sin ( 2 x )}{5}

= sin^{2} (x) e^{x} - (- \frac{2 e ^{x} cos ( 2 x )}{5} + \frac{e ^{x} sin ( 2 x )}{5})

Simplify

sin^{2} (x) e^{x} - (- \frac{2 e ^{x} cos ( 2 x )}{5} + \frac{e ^{x} sin ( 2 x )}{5}) : e^{x} sin^{2} (x) + \frac{2}{5} e^{x} cos (2 x) - \frac{1}{5} e^{x} sin (2 x)

= e^{x} sin^{2} (x) + \frac{2}{5} e^{x} cos (2 x) - \frac{1}{5} e^{x} sin (2 x)

Add a constant to the solution

= e^{x} sin^{2} (x) + \frac{2}{5} e^{x} cos (2 x) - \frac{1}{5} e^{x} sin (2 x) + C

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What is the integral of e^x(sin(x))^2 ?
The integral of e^x(sin(x))^2 is e^xsin^2(x)+2/5 e^xcos(2x)-1/5 e^xsin(2x)+C