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

The integral from-pi to pi of |x|sin(nx) is 0

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integral from-pi to pi of |x|sin(nx)

\int_{- π}^{π} ∣ x ∣ sin (n x) d x

0

Solution steps

\int_{- π}^{π} ∣ x ∣ sin (n x) d x

Eliminate Absolutes

= \int_{- π}^{0} (- x) sin (n x) d x + \int_{0}^{π} x sin (n x) d x

\int_{- π}^{0} (- x) sin (n x) d x = \frac{π ( - 1 ) ^{n}}{n}

\int_{0}^{π} x sin (n x) d x = - \frac{π ( - 1 ) ^{n}}{n}

= \frac{π ( - 1 ) ^{n}}{n} - \frac{π ( - 1 ) ^{n}}{n}

Simplify

= 0

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

What is the integral from-pi to pi of |x|sin(nx) ?
The integral from-pi to pi of |x|sin(nx) is 0