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

The integral of e^x(-sin(x)) is 1/2 e^xcos(x)-1/2 e^xsin(x)+C

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integral of e^x(-sin(x))

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

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

Solution steps

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

Simplify

= \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

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

Apply Integration By Parts

= - (- 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

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

Apply Integration By Parts

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

Therefore

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

Isolate

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

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

Simplify

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

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

Add a constant to the solution

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

What is the integral of e^x(-sin(x)) ?
The integral of e^x(-sin(x)) is 1/2 e^xcos(x)-1/2 e^xsin(x)+C