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

The integral from 0 to pi/3 of xcos(3x) is -2/9

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integral from 0 to pi/3 of xcos(3x)

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

- \frac{2}{9}

Decimal

- 0.22222 \dots

Solution steps

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

Apply u-substitution

: \int_{0}^{π} \frac{u cos ( u )}{9} d u

= \int_{0}^{π} \frac{u cos ( u )}{9} d u

Take the constant out:

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

= \frac{1}{9} \cdot \int_{0}^{π} u cos (u) d u

Apply Integration By Parts

: [u sin (u) - \int sin (u) d u]_{0}^{π}

= \frac{1}{9} [u sin (u) - \int sin (u) d u]_{0}^{π}

\int sin (u) d u = - cos (u)

= \frac{1}{9} [u sin (u) - (- cos (u))]_{0}^{π}

Simplify

= \frac{1}{9} [u sin (u) + cos (u)]_{0}^{π}

Compute the boundaries:

- 2

= \frac{1}{9} (- 2)

Simplify

= - \frac{2}{9}

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What is the integral from 0 to pi/3 of xcos(3x) ?
The integral from 0 to pi/3 of xcos(3x) is -2/9