What is the integral of (e^xsin(x))/2 ?

The integral of (e^xsin(x))/2 is 1/4 (-e^xcos(x)+e^xsin(x))+C

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integral of (e^xsin(x))/2

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

\frac{1}{4} (- e^{x} cos (x) + e^{x} sin (x)) + C

Solution steps

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

Take the constant out:

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

= \frac{1}{2} \cdot \int e^{x} sin (x) d x

Apply Integration By Parts

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

= \frac{1}{2} (- 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

= \frac{1}{2} (- 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

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

Therefore

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

Isolate

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

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

Simplify

\frac{1}{2} (- \frac{e ^{x} cos ( x )}{2} + \frac{e ^{x} sin ( x )}{2}) : \frac{1}{4} (- e^{x} cos (x) + e^{x} sin (x))

= \frac{1}{4} (- e^{x} cos (x) + e^{x} sin (x))

Add a constant to the solution

= \frac{1}{4} (- e^{x} cos (x) + e^{x} sin (x)) + C

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What is the integral of (e^xsin(x))/2 ?
The integral of (e^xsin(x))/2 is 1/4 (-e^xcos(x)+e^xsin(x))+C