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

The integral of e^x(sin(x)+cos(x)) is e^xsin(x)+C

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integral of e^x(sin(x)+cos(x))

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

e^{x} sin (x) + C

Solution steps

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

Expand

e^{x} (sin (x) + cos (x)) : e^{x} sin (x) + e^{x} cos (x)

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

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

Simplify

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

= e^{x} sin (x)

Add a constant to the solution

= e^{x} sin (x) + C

\frac{\partial}{\partial y} (x - 4 y - 9)

\int (4 + 3 x)^{4} d x

n = 1 \sum \infty \frac{n - 1}{2 n + 1}

cos (x^{- 9})

\frac{d}{d x} (3 - \frac{7}{2 x ^{\frac{1}{8}}})

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