What is the integral from 0 to pi/3 of xsin(12x) ?

The integral from 0 to pi/3 of xsin(12x) is -pi/(36)

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integral from 0 to pi/3 of xsin(12x)

\int_{0}^{\frac{π}{3}} x sin (12 x) d x

- \frac{π}{36}

Decimal Notation

- 0.08726 \dots

Solution steps

\int_{0}^{\frac{π}{3}} x sin (12 x) d x

Apply Integration By Parts

: [\frac{1}{12} (- x cos (12 x) - 12 \cdot \int - \frac{1}{12} cos (12 x) d x)]_{0}^{\frac{π}{3}}

= [\frac{1}{12} (- x cos (12 x) - 12 \cdot \int - \frac{1}{12} cos (12 x) d x)]_{0}^{\frac{π}{3}}

\int - \frac{1}{12} cos (12 x) d x = - \frac{1}{144} sin (12 x)

= [\frac{1}{12} (- x cos (12 x) - 12 (- \frac{1}{144} sin (12 x)))]_{0}^{\frac{π}{3}}

Simplify

[\frac{1}{12} (- x cos (12 x) - 12 (- \frac{1}{144} sin (12 x)))]_{0}^{\frac{π}{3}} : [\frac{1}{12} (- x cos (12 x) + \frac{1}{12} sin (12 x))]_{0}^{\frac{π}{3}}

= [\frac{1}{12} (- x cos (12 x) + \frac{1}{12} sin (12 x))]_{0}^{\frac{π}{3}}

Compute the boundaries:

- \frac{π}{36}

= - \frac{π}{36}

\int x^{\frac{2}{3}} d x

f (x) = \frac{sin ^{2} ( x )}{x ^{2}}

\frac{d y}{d x} = (y - 1)^{- 3}

\int \frac{sin ( θ )}{cos ^{2} ( θ ) - cos ( θ ) - 2} d θ

y (θ) = cos (θ^{2})

What is the integral from 0 to pi/3 of xsin(12x) ?
The integral from 0 to pi/3 of xsin(12x) is -pi/(36)