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{\partial}{\partial x} (y^{2} - 4 x^{2} + 7 y + 5 x + 13)

y^{^{'}} + 5 (cos (5 x)) y = - 8 cos (5 x)

y^{2} = 4 x, y^{2} = 8 x - 4, x

x \to 42 lim (\frac{x ^{2}}{6} - \frac{7}{x})

\frac{1 + 10 s}{s ^{2} + 10.1 s + 1}