What is the integral from pi/6 to pi/2 of xcos(x) ?

The integral from pi/6 to pi/2 of xcos(x) is (5pi-6sqrt(3))/(12)

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integral from pi/6 to pi/2 of xcos(x)

\int_{\frac{π}{6}}^{\frac{π}{2}} x cos (x) d x

\frac{5 π - 6 3}{12}

Decimal Notation

0.44297 \dots

Solution steps

\int_{\frac{π}{6}}^{\frac{π}{2}} x cos (x) d x

Apply Integration By Parts

: [x sin (x) - \int sin (x) d x]_{\frac{π}{6}}^{\frac{π}{2}}

= [x sin (x) - \int sin (x) d x]_{\frac{π}{6}}^{\frac{π}{2}}

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

= [x sin (x) - (- cos (x))]_{\frac{π}{6}}^{\frac{π}{2}}

Simplify

= [x sin (x) + cos (x)]_{\frac{π}{6}}^{\frac{π}{2}}

Compute the boundaries:

\frac{5 π - 6 3}{12}

= \frac{5 π - 6 3}{12}

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What is the integral from pi/6 to pi/2 of xcos(x) ?
The integral from pi/6 to pi/2 of xcos(x) is (5pi-6sqrt(3))/(12)