What is the integral from-pi to pi of x*sin(x) ?

The integral from-pi to pi of x*sin(x) is 2pi

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integral from-pi to pi of x*sin(x)

\int_{- π}^{π} x \cdot sin (x) d x

2 π

Decimal

6.28318 \dots

Solution steps

\int_{- π}^{π} x sin (x) d x

Apply Integration By Parts

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

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

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

= [- x cos (x) - (- sin (x))]_{- π}^{π}

Simplify

= [- x cos (x) + sin (x)]_{- π}^{π}

Compute the boundaries:

2 π

= 2 π

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

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What is the integral from-pi to pi of x*sin(x) ?
The integral from-pi to pi of x*sin(x) is 2pi