Check the solution to "Solve the integral ∫sin⁡xexdx"

Mylo O'Moore

Answered question

2021-11-07

Solve the integral

\int \frac{\sin x}{e^{x}} d x

Talisha

Skilled2021-11-08Added 93 answers

Step 1
The given integral is

\int \frac{\sin x}{e^{x}} d x

The above integral can be represented by I as,

I = \int \sin x e^{- x} d x

Solving the above integral as,

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

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

I = - \sin x e^{- x} - \cos x e^{- x} - \int \sin x e^{- x} d x

I = - \sin x e^{- x} - \cos x e^{- x} - I + c

Step 2
It can further simplified as,

2 I = - \sin x e^{- x} - \cos x e^{- x} + c

I = \frac{- \sin x e^{- x} - \cos x e^{- x}}{2} + c

Hence, the value of the given integral is

\frac{- \sin x e^{- x} - \cos x e^{- x}}{2} + c

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