What is the integral of 2xsin(3xy) ?

The integral of 2xsin(3xy) is 2(-1/(3y)xcos(3yx)+1/(9y^2)sin(3yx))+C

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integral of 2xsin(3xy)

\int 2 x sin (3 x y) d x

2 (- \frac{1}{3 y} x cos (3 y x) + \frac{1}{9 y ^{2}} sin (3 y x)) + C

Solution steps

\int 2 x sin (3 x y) d x

Take the constant out:

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

= 2 \cdot \int sin (3 y x) x d x

Apply Integration By Parts

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

= 2 (- \frac{1}{3 y} x cos (3 y x) - \int - \frac{1}{3 y} cos (3 y x) d x)

\int - \frac{1}{3 y} cos (3 y x) d x = - \frac{1}{9 y ^{2}} sin (3 y x)

= 2 (- \frac{1}{3 y} x cos (3 y x) - (- \frac{1}{9 y ^{2}} sin (3 y x)))

Simplify

= 2 (- \frac{1}{3 y} x cos (3 y x) + \frac{1}{9 y ^{2}} sin (3 y x))

Add a constant to the solution

= 2 (- \frac{1}{3 y} x cos (3 y x) + \frac{1}{9 y ^{2}} sin (3 y x)) + C

x \to - 4 lim (x^{3} + x^{2})

\int_{2}^{5} f (x) d x

x \to \infty lim (- (\frac{1}{x}))

\frac{\partial}{\partial x} (e^{- 2 x^{2} - y^{2}})

What is the integral of 2xsin(3xy) ?
The integral of 2xsin(3xy) is 2(-1/(3y)xcos(3yx)+1/(9y^2)sin(3yx))+C