What is the integral of 3sin(x)e^x ?

The integral of 3sin(x)e^x is 3(-(e^xcos(x))/2+(e^xsin(x))/2)+C

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integral of 3sin(x)e^x

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

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

Solution steps

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

Take the constant out:

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

= 3 \cdot \int sin (x) e^{x} d x

Apply Integration By Parts

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

= 3 (- 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

= 3 (- 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

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

Therefore

3 \cdot \int e^{x} sin (x) d x = 3 (- e^{x} cos (x) - (- (e^{x} sin (x) - \int e^{x} sin (x) d x)))

Isolate

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

= 3 (- \frac{e ^{x} cos ( x )}{2} + \frac{e ^{x} sin ( x )}{2})

Add a constant to the solution

= 3 (- \frac{e ^{x} cos ( x )}{2} + \frac{e ^{x} sin ( x )}{2}) + C

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What is the integral of 3sin(x)e^x ?
The integral of 3sin(x)e^x is 3(-(e^xcos(x))/2+(e^xsin(x))/2)+C