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

The integral of 3xsin(2x) is 3(-1/2 xcos(2x)+1/4 sin(2x))+C

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integral of 3xsin(2x)

\int 3 x sin (2 x) d x

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

Solution steps

\int 3 x sin (2 x) d x

Take the constant out:

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

= 3 \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

= 3 (- \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)

= 3 (- \frac{1}{2} x cos (2 x) - (- \frac{1}{4} sin (2 x)))

Simplify

= 3 (- \frac{1}{2} x cos (2 x) + \frac{1}{4} sin (2 x))

Add a constant to the solution

= 3 (- \frac{1}{2} x cos (2 x) + \frac{1}{4} sin (2 x)) + C

\frac{d}{d x} (\frac{2 x cos ( x ^{2} )}{1 + sin ( x ^{2} )})

f^{^{'}} (x) = x^{3}

\frac{1}{x + 5}

\frac{d}{d x} (x + i y (x))

(x + y)^{2} d x + (2 x y + x^{2} - 5) d y = 0, y (1) = 1

What is the integral of 3xsin(2x) ?
The integral of 3xsin(2x) is 3(-1/2 xcos(2x)+1/4 sin(2x))+C