What is the integral of sin(uv) ?

The integral of sin(uv) is -1/v cos(vu)+C

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integral of sin(uv)

\int sin (uv) d u

- \frac{1}{v} cos (vu) + C

Solution steps

\int sin (uv) d u

Apply u-substitution

: \int \frac{sin ( w )}{v} d w

= \int \frac{sin ( w )}{v} d w

Take the constant out:

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

= \frac{1}{v} \cdot \int sin (w) d w

Use the common integral:

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

= \frac{1}{v} (- cos (w))

Substitute back

w = vu

= \frac{1}{v} (- cos (vu))

Simplify

= - \frac{1}{v} cos (vu)

Add a constant to the solution

= - \frac{1}{v} cos (vu) + C

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