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

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

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

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

e^{x} sin (x) + C

Solution steps

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

Expand

(sin (x) + cos (x)) e^{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{d}{d x} (e^{x^{3} + 2})

2 x^{2} - 5 x

f (x) = (2 x - 9)^{4} (x^{2} + x + 1)^{5}

y = x^{2} - 2 x, (5, 15)

\frac{1}{( s + 2 ) ^{2}}

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