What is the integral of 4xsin(2x) ?

The integral of 4xsin(2x) is -2xcos(2x)+sin(2x)+C

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integral of 4xsin(2x)

\int 4 x sin (2 x) d x

- 2 x cos (2 x) + sin (2 x) + C

Solution steps

\int 4 x sin (2 x) d x

Take the constant out:

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

= 4 \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

= 4 (- \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)

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

Simplify

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

= - 2 x cos (2 x) + sin (2 x)

Add a constant to the solution

= - 2 x cos (2 x) + sin (2 x) + C

(- 3, - 4), (0, 2)

\int \frac{1}{10 x ^{3}} d x

\int x^{2} 6 + x^{3} d x

\frac{d}{d r} (\frac{2}{r ^{3}})

y^{^{''}} + 10 y^{^{'}} + 41 y = 0

What is the integral of 4xsin(2x) ?
The integral of 4xsin(2x) is -2xcos(2x)+sin(2x)+C