What is the integral of e^ycos(y) ?

The integral of e^ycos(y) is (e^ysin(y))/2+(e^ycos(y))/2+C

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integral of e^ycos(y)

\int e^{y} cos (y) d y

\frac{e ^{y} sin ( y )}{2} + \frac{e ^{y} cos ( y )}{2} + C

Solution steps

\int e^{y} cos (y) d y

Apply Integration By Parts

: e^{y} sin (y) - \int e^{y} sin (y) d y

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Therefore

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

Isolate

\int e^{y} cos (y) d y

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

Add a constant to the solution

= \frac{e ^{y} sin ( y )}{2} + \frac{e ^{y} cos ( y )}{2} + C

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What is the integral of e^ycos(y) ?
The integral of e^ycos(y) is (e^ysin(y))/2+(e^ycos(y))/2+C