What is the integral of (sin(xy)) ?

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

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integral of (sin(xy))

\int (sin (x y)) d y

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

Solution steps

\int sin (x y) d y

Apply u-substitution

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

= \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

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

Use the common integral:

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

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

Substitute back

u = x y

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

Simplify

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

Add a constant to the solution

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

\frac{d ^{2} x}{d t ^{2}} + 6 \frac{d x}{d t} + 8 x = 5 sin (3 t)

\frac{d ^{2}}{d x ^{2}} (x^{\frac{9}{2}} e^{x})

x \to \infty lim (- 2 (x))

n = 1 \sum \infty \frac{9 n}{10}

(- 10, - 3), (4, - 3)