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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

Simplify

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

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

Add a constant to the solution

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

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