What is the integral of-ysin(xy) ?

The integral of-ysin(xy) is cos(yx)+C

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integral of-ysin(xy)

\int - y sin (x y) d x

cos (y x) + C

Solution steps

\int - y sin (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 sin (y x) d x

Apply u-substitution

: \int \frac{sin ( u )}{y} d u

= - y \cdot \int \frac{sin ( 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 sin (u) d u

Use the common integral:

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

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

Substitute back

u = y x

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

Simplify

- y \frac{1}{y} (- cos (y x)) : cos (y x)

= cos (y x)

Add a constant to the solution

= cos (y x) + C

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