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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

\int (x^{3} - x) e^{x} d x

\int \frac{x - 3}{x ^{2} - 2 x + 1} d x

\int x^{2} 6 - x d x

\int (x^{4} - \frac{6}{x ^{4}}) d x

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

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