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

The integral of sin(sqrt(xy)) is 2/y (-cos(sqrt(yx))sqrt(yx)+sin(sqrt(yx)))+C

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integral of sin(sqrt(xy))

\int sin (x y) d x

\frac{2}{y} (- cos (y x) y x + sin (y x)) + C

Solution steps

\int sin (x y) d x

Apply u-substitution

: \int \frac{2 u sin ( u )}{y} d u

= \int \frac{2 u sin ( u )}{y} d u

Take the constant out:

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

= \frac{2}{y} \cdot \int u sin (u) d u

Apply Integration By Parts

: - u cos (u) - \int - cos (u) d u

= \frac{2}{y} (- u cos (u) - \int - cos (u) d u)

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

= \frac{2}{y} (- u cos (u) - (- sin (u)))

Substitute back

u = y x

= \frac{2}{y} (- y x cos (y x) - (- sin (y x)))

Simplify

= \frac{2}{y} (- cos (y x) y x + sin (y x))

Add a constant to the solution

= \frac{2}{y} (- cos (y x) y x + sin (y x)) + C

\int e^{3 - 8 x} d x

\int - 4 sin (t) d t

\int e^{- 2 x} (4 - x) d x

\int x lo g_{3} (x) d x

x^{\frac{2}{3}}

What is the integral of sin(sqrt(xy)) ?
The integral of sin(sqrt(xy)) is 2/y (-cos(sqrt(yx))sqrt(yx)+sin(sqrt(yx)))+C