What is the integral of 3ysin(xy) ?

The integral of 3ysin(xy) is -3cos(yx)+C

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integral of 3ysin(xy)

\int 3 y sin (x y) d x

- 3 cos (y x) + C

Solution steps

\int 3 y sin (x y) d x

Take the constant out:

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

= 3 y \cdot \int sin (y x) d x

Apply u-substitution

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

= 3 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

= 3 y \frac{1}{y} \cdot \int sin (u) d u

Use the common integral:

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

= 3 y \frac{1}{y} (- cos (u))

Substitute back

u = y x

= 3 y \frac{1}{y} (- cos (y x))

Simplify

3 y \frac{1}{y} (- cos (y x)) : - 3 cos (y x)

= - 3 cos (y x)

Add a constant to the solution

= - 3 cos (y x) + C

\frac{d y}{d x} = 7 x^{5} + 6

f (x) = 7 e^{- x} - 4 e^{- 7 x}

\frac{\partial}{\partial x} (3 sec^{2} (3 x - 2 y + 1))

\frac{d}{d x} ((2 x + 3)^{5} + 10 x (2 x + 3)^{4})

21 x y^{^{'}} + y = 462 x