What is the integral of ycos(xy) ?

The integral of ycos(xy) is sin(yx)+C

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integral of ycos(xy)

\int y cos (x y) d x

sin (y x) + C

Solution steps

\int 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) d x

Apply u-substitution

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

Take the constant out:

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

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

Use the common integral:

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

= y \frac{1}{y} sin (u)

Substitute back

u = y x

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

Simplify

y \frac{1}{y} sin (y x) : sin (y x)

= sin (y x)

Add a constant to the solution

= sin (y x) + C