What is the integral of ye^{xy}cos(2x) ?

The integral of ye^{xy}cos(2x) is ((2e^{yx}sin(2x))/(4+y^2)+(ye^{yx}cos(2x))/(4+y^2))y+C

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integral of ye^{xy}cos(2x)

\int y e^{x y} cos (2 x) d x

(\frac{2 e ^{y x} sin ( 2 x )}{4 + y ^{2}} + \frac{y e ^{y x} cos ( 2 x )}{4 + y ^{2}}) y + C

Solution steps

\int y e^{x y} cos (2 x) d x

Take the constant out:

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

= y \cdot \int cos (2 x) e^{y x} d x

Apply Integration By Parts

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

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

Take the constant out:

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

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

Simplify

y \frac{1}{2} : \frac{y}{2}

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Simplify

- \frac{1}{2} y : - \frac{y}{2}

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

Therefore

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

Isolate

\int e^{y x} cos (2 x) d x

= (\frac{2 e ^{y x} sin ( 2 x )}{4 + y ^{2}} + \frac{y e ^{y x} cos ( 2 x )}{4 + y ^{2}}) y

Add a constant to the solution

= (\frac{2 e ^{y x} sin ( 2 x )}{4 + y ^{2}} + \frac{y e ^{y x} cos ( 2 x )}{4 + y ^{2}}) y + C

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\frac{d}{d x} ((sin (5 x)) (arccsc (2) x^{3}))

x y^{^{'}} = y + 2 x^{3} sin^{2} (\frac{y}{x})

\frac{d}{d x} (x^{2} + 100)

What is the integral of ye^{xy}cos(2x) ?
The integral of ye^{xy}cos(2x) is ((2e^{yx}sin(2x))/(4+y^2)+(ye^{yx}cos(2x))/(4+y^2))y+C