What is the integral of sqrt(x)sin(sqrt(x)y) ?

The integral of sqrt(x)sin(sqrt(x)y) is -cos(sqrt(x)y)+C

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integral of sqrt(x)sin(sqrt(x)y)

\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

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

= 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

x \to - 1 lim (2 x - 3)

x \to \infty lim ((1 + \frac{8}{x})^{\frac{x}{9}})

x \to 0 lim (ln (\frac{1}{x}))

\frac{4 s e ^{- s}}{2 s ^{2} + 32}

\int \frac{x ^{2} - x + 12}{x ^{3} + 2 x} d x

What is the integral of sqrt(x)sin(sqrt(x)y) ?
The integral of sqrt(x)sin(sqrt(x)y) is -cos(sqrt(x)y)+C