What is the integral from 0 to pi/6 of xcos(x+pi/3) ?

The integral from 0 to pi/6 of xcos(x+pi/3) is pi/6-1/2

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integral from 0 to pi/6 of xcos(x+pi/3)

\int_{0}^{\frac{π}{6}} x cos (x + \frac{π}{3}) d x

\frac{π}{6} - \frac{1}{2}

Decimal

0.02359 \dots

Solution steps

\int_{0}^{\frac{π}{6}} x cos (x + \frac{π}{3}) d x

Apply Integration By Parts

= [x sin (x + \frac{π}{3}) - \int sin (x + \frac{π}{3}) d x]_{0}^{\frac{π}{6}}

\int sin (x + \frac{π}{3}) d x = - cos (x + \frac{π}{3})

= [x sin (x + \frac{π}{3}) - (- cos (x + \frac{π}{3}))]_{0}^{\frac{π}{6}}

Simplify

= [x sin (x + \frac{π}{3}) + cos (x + \frac{π}{3})]_{0}^{\frac{π}{6}}

Compute the boundaries:

\frac{π}{6} - \frac{1}{2}

= \frac{π}{6} - \frac{1}{2}

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What is the integral from 0 to pi/6 of xcos(x+pi/3) ?
The integral from 0 to pi/6 of xcos(x+pi/3) is pi/6-1/2