What is the integral of-xsin(xy) ?

The integral of-xsin(xy) is cos(xy)+C

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integral of-xsin(xy)

\int - x sin (x y) d y

cos (x y) + C

Solution steps

\int - x sin (x y) d y

Take the constant out:

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

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

Apply u-substitution

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

u = x y

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

Simplify

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

= cos (x y)

Add a constant to the solution

= cos (x y) + C