What is the integral of sin(xy) ?

The integral of sin(xy) is -1/y cos(yx)+C

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integral of sin(xy)

\int sin (x y) d x

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

Solution steps

\int sin (x y) d x

Apply u-substitution

= \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

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

Use the common integral:

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

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

Substitute back

u = y x

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

Simplify

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

Add a constant to the solution

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