What is the integral of xycos(xy) ?

The integral of xycos(xy) is xsin(yx)+1/y cos(yx)+C

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integral of xycos(xy)

\int x y cos (x y) d x

x sin (y x) + \frac{1}{y} cos (y x) + C

Solution steps

\int x y cos (x y) d x

Take the constant out:

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

= y \cdot \int cos (y x) x d x

Apply u-substitution

: \int \frac{u cos ( u )}{y ^{2}} d u

= y \cdot \int \frac{u cos ( u )}{y ^{2}} d u

Take the constant out:

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

= y \frac{1}{y ^{2}} \cdot \int u cos (u) d u

Apply Integration By Parts

: u sin (u) - \int sin (u) d u

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

\int sin (u) d u = - cos (u)

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

Substitute back

u = y x

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

Simplify

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

= x sin (y x) + \frac{1}{y} cos (y x)

Add a constant to the solution

= x sin (y x) + \frac{1}{y} cos (y x) + C

\frac{d y}{d x} = (y - y^{2})

\int e^{\frac{2}{t}} d t

\int sin^{3} (5 x) cos (5 x) d x

\int_{1}^{2} x 3^{(x^{2})} d x

(x^{2} + y^{2}) d x + x y d y = 0

What is the integral of xycos(xy) ?
The integral of xycos(xy) is xsin(yx)+1/y cos(yx)+C