What is the integral from 0 to pi/3 of 2xcos(x) ?

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

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integral from 0 to pi/3 of 2xcos(x)

\int_{0}^{\frac{π}{3}} 2 x cos (x) d x

- 1 + \frac{π}{3}

Decimal Notation

0.81379 \dots

Solution steps

\int_{0}^{\frac{π}{3}} 2 x cos (x) d x

Take the constant out:

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

= 2 \cdot \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}}

= 2 [x sin (x) - \int sin (x) d x]_{0}^{\frac{π}{3}}

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

= 2 [x sin (x) - (- cos (x))]_{0}^{\frac{π}{3}}

Simplify

= 2 [x sin (x) + cos (x)]_{0}^{\frac{π}{3}}

Compute the boundaries:

- \frac{1}{2} + \frac{π}{2 3}

= 2 (- \frac{1}{2} + \frac{π}{2 3})

Simplify

= - 1 + \frac{π}{3}

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