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

The integral of 2sin(x)e^x is -e^xcos(x)+e^xsin(x)+C

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integral of 2sin(x)e^x

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

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

Solution steps

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

Apply Integration By Parts

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

Apply Integration By Parts

= 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

What is the integral of 2sin(x)e^x ?
The integral of 2sin(x)e^x is -e^xcos(x)+e^xsin(x)+C