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

The integral of sin^2(x)e^{-x} is -e^{-x}sin^2(x)-2/5 e^{-x}cos(2x)-1/5 e^{-x}sin(2x)+C

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integral of sin^2(x)e^{-x}

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

Apply Integration By Parts

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

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

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

Simplify

= - 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 sin^2(x)e^{-x} ?
The integral of sin^2(x)e^{-x} is -e^{-x}sin^2(x)-2/5 e^{-x}cos(2x)-1/5 e^{-x}sin(2x)+C