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

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

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

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

- e^{- x} cos (x) - e^{- x} sin (x) + C

Solution steps

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Therefore

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

Isolate

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

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

Simplify

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

= - e^{- x} cos (x) - e^{- x} sin (x)

Add a constant to the solution

= - e^{- x} cos (x) - e^{- x} sin (x) + C

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