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

The integral from 0 to pi/3 of xsin(x) is (sqrt(3))/2-pi/6

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

\int_{0}^{\frac{π}{3}} x sin (x) d x

\frac{3}{2} - \frac{π}{6}

Decimal

0.34242 \dots

Solution steps

\int_{0}^{\frac{π}{3}} x sin (x) d x

Apply Integration By Parts

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

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

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

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

Simplify

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

Compute the boundaries:

\frac{3}{2} - \frac{π}{6}

= \frac{3}{2} - \frac{π}{6}

\int_{6}^{8} \frac{x}{50} d x

\int_{4}^{16} \frac{x}{2} - x d x

\int_{0}^{x} sin (t^{2}) d t

\int_{0}^{1} 2 x - 2 d x

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

What is the integral from 0 to pi/3 of xsin(x) ?
The integral from 0 to pi/3 of xsin(x) is (sqrt(3))/2-pi/6