What is the integral from 0 to pi/4 of xsin(16x) ?

The integral from 0 to pi/4 of xsin(16x) is -pi/(64)

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integral from 0 to pi/4 of xsin(16x)

\int_{0}^{\frac{π}{4}} x sin (16 x) d x

- \frac{π}{64}

Decimal

- 0.04908 \dots

Solution steps

\int_{0}^{\frac{π}{4}} x sin (16 x) d x

Apply Integration By Parts

: [\frac{1}{16} (- x cos (16 x) - 16 \cdot \int - \frac{1}{16} cos (16 x) d x)]_{0}^{\frac{π}{4}}

= [\frac{1}{16} (- x cos (16 x) - 16 \cdot \int - \frac{1}{16} cos (16 x) d x)]_{0}^{\frac{π}{4}}

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

= [\frac{1}{16} (- x cos (16 x) - 16 (- \frac{1}{256} sin (16 x)))]_{0}^{\frac{π}{4}}

Simplify

[\frac{1}{16} (- x cos (16 x) - 16 (- \frac{1}{256} sin (16 x)))]_{0}^{\frac{π}{4}} : [\frac{1}{16} (- x cos (16 x) + \frac{1}{16} sin (16 x))]_{0}^{\frac{π}{4}}

= [\frac{1}{16} (- x cos (16 x) + \frac{1}{16} sin (16 x))]_{0}^{\frac{π}{4}}

Compute the boundaries:

- \frac{π}{64}

= - \frac{π}{64}

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4 t \frac{d y}{d t} - 5 y = t

What is the integral from 0 to pi/4 of xsin(16x) ?
The integral from 0 to pi/4 of xsin(16x) is -pi/(64)