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

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

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

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

33 - π

Decimal Notation

2.05455 \dots

Solution steps

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

Take the constant out:

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

= 6 \cdot \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}}

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

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

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

Simplify

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

Compute the boundaries:

\frac{3}{2} - \frac{π}{6}

= 6 (\frac{3}{2} - \frac{π}{6})

Simplify

= 33 - π

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