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

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

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integral of 4xsin(4x)

\int 4 x sin (4 x) d x

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

Solution steps

\int 4 x sin (4 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 (4 x) d x

Apply Integration By Parts

: - \frac{1}{4} x cos (4 x) - \int - \frac{1}{4} cos (4 x) d x

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

\int - \frac{1}{4} cos (4 x) d x = - \frac{1}{16} sin (4 x)

= 4 (- \frac{1}{4} x cos (4 x) - (- \frac{1}{16} sin (4 x)))

Simplify

4 (- \frac{1}{4} x cos (4 x) - (- \frac{1}{16} sin (4 x))) : - x cos (4 x) + \frac{1}{4} sin (4 x)

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

Add a constant to the solution

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

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