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

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

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

\int_{0}^{\frac{π}{3}} x cos (x) d x

- \frac{1}{2} + \frac{π}{2 3}

Decimal

0.40689 \dots

Solution steps

\int_{0}^{\frac{π}{3}} x cos (x) d x

Apply Integration By Parts

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

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

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

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

Simplify

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

Compute the boundaries:

- \frac{1}{2} + \frac{π}{2 3}

= - \frac{1}{2} + \frac{π}{2 3}

\int_{0}^{2 π} - cos (θ) d θ

\int_{0}^{1} x^{2} (1 + x^{3})^{2} d x

\int_{0}^{4} \frac{1}{2} x + 2 d x

\int_{0}^{2} π^{8} x d x

\int_{0}^{1} t 3 7^{t^{2}} d t

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