What is the integral of-1/2 e^ysin(y) ?

The integral of-1/2 e^ysin(y) is -1/4 (-e^ycos(y)+e^ysin(y))+C

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integral of-1/2 e^ysin(y)

\int - \frac{1}{2} e^{y} sin (y) d y

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

Solution steps

\int - \frac{1}{2} e^{y} sin (y) d y

Take the constant out:

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

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

Apply Integration By Parts

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

= - \frac{1}{2} (- 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

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

Apply Integration By Parts

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

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

Therefore

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

Isolate

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

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

Simplify

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

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

Add a constant to the solution

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

\frac{d}{d x} (\frac{686}{( 1 + 7 x ) ^{3}})

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\int \frac{x ^{2}}{16 - 25 x ^{2}} d x

\int \frac{( 16 x )}{( x ^{2} - 9 ) ^{5}} d x

What is the integral of-1/2 e^ysin(y) ?
The integral of-1/2 e^ysin(y) is -1/4 (-e^ycos(y)+e^ysin(y))+C