What is the integral of ysin(xy) ?

The integral of ysin(xy) is -cos(yx)+C

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integral of ysin(xy)

\int y sin (x y) d x

- cos (y x) + C

Solution steps

\int y sin (x y) d x

Take the constant out:

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

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

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

u = y x

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

Simplify

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

= - cos (y x)

Add a constant to the solution

= - cos (y x) + C

\frac{d}{d x} (- 4 x^{10})

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\int (x - 4) sin (π x) d x

\int \frac{4 x}{( 1 + x ^{2} )} d x

3 x y^{^{'}} - 2 y = x^{3} y^{- 2}