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

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

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

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

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

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Apply Integration By Parts

: - \frac{1}{2} e^{- 2 x} sin (x) - \int - \frac{1}{2} e^{- 2 x} cos (x) d x

= 2 (- \frac{1}{2} e^{- 2 x} sin (x) - \int - \frac{1}{2} e^{- 2 x} cos (x) d x)

\int - \frac{1}{2} e^{- 2 x} cos (x) d x = - \frac{1}{10} (e^{- 2 x} sin (x) - 2 e^{- 2 x} cos (x))

= 2 (- \frac{1}{2} e^{- 2 x} sin (x) - (- \frac{1}{10} (e^{- 2 x} sin (x) - 2 e^{- 2 x} cos (x))))

Simplify

2 (- \frac{1}{2} e^{- 2 x} sin (x) - (- \frac{1}{10} (e^{- 2 x} sin (x) - 2 e^{- 2 x} cos (x)))) : 2 (- \frac{2}{5} e^{- 2 x} sin (x) - \frac{1}{5} e^{- 2 x} cos (x))

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

Add a constant to the solution

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

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