What is the integral of 2xsin(2x) ?

The integral of 2xsin(2x) is -xcos(2x)+1/2 sin(2x)+C

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integral of 2xsin(2x)

\int 2 x sin (2 x) d x

- x cos (2 x) + \frac{1}{2} sin (2 x) + C

Solution steps

\int 2 x sin (2 x) d x

Take the constant out:

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

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

Apply Integration By Parts

: - \frac{1}{2} x cos (2 x) - \int - \frac{1}{2} cos (2 x) d x

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

\int - \frac{1}{2} cos (2 x) d x = - \frac{1}{4} sin (2 x)

= 2 (- \frac{1}{2} x cos (2 x) - (- \frac{1}{4} sin (2 x)))

Simplify

2 (- \frac{1}{2} x cos (2 x) - (- \frac{1}{4} sin (2 x))) : - x cos (2 x) + \frac{1}{2} sin (2 x)

= - x cos (2 x) + \frac{1}{2} sin (2 x)

Add a constant to the solution

= - x cos (2 x) + \frac{1}{2} sin (2 x) + C

\int - k x e^{- k x} d x

\frac{\partial}{\partial x} (cos (x + y))

\int (2 x - 1) d x

- \frac{3}{x}

\frac{d}{d x} (a cos^{2} (x))

What is the integral of 2xsin(2x) ?
The integral of 2xsin(2x) is -xcos(2x)+1/2 sin(2x)+C