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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

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