What is the integral of xsin^2(8x) ?

The integral of xsin^2(8x) is 1/512 (128x^2-16xsin(16x)-cos(16x))+C

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integral of xsin^2(8x)

\int x sin^{2} (8 x) d x

\frac{1}{512} (128 x^{2} - 16 x sin (16 x) - cos (16 x)) + C

Solution steps

\int x sin^{2} (8 x) d x

Rewrite using trig identities

= \int \frac{1}{2} x (1 - cos (16 x)) d x

Take the constant out:

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

= \frac{1}{2} \cdot \int x (1 - cos (16 x)) d x

Apply Integration By Parts

: x (x - \frac{1}{16} sin (16 x)) - \int x - \frac{1}{16} sin (16 x) d x

= \frac{1}{2} (x (x - \frac{1}{16} sin (16 x)) - \int x - \frac{1}{16} sin (16 x) d x)

\int x - \frac{1}{16} sin (16 x) d x = \frac{x ^{2}}{2} + \frac{1}{256} cos (16 x)

= \frac{1}{2} (x (x - \frac{1}{16} sin (16 x)) - (\frac{x ^{2}}{2} + \frac{1}{256} cos (16 x)))

Simplify

\frac{1}{2} (x (x - \frac{1}{16} sin (16 x)) - (\frac{x ^{2}}{2} + \frac{1}{256} cos (16 x))) : \frac{1}{512} (128 x^{2} - 16 x sin (16 x) - cos (16 x))

= \frac{1}{512} (128 x^{2} - 16 x sin (16 x) - cos (16 x))

Add a constant to the solution

= \frac{1}{512} (128 x^{2} - 16 x sin (16 x) - cos (16 x)) + C

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What is the integral of xsin^2(8x) ?
The integral of xsin^2(8x) is 1/512 (128x^2-16xsin(16x)-cos(16x))+C