What is the integral from-pi to pi of sin(m)xsin(n)x ?

The integral from-pi to pi of sin(m)xsin(n)x is (2pi^3)/3 sin(m)sin(n)

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integral from-pi to pi of sin(m)xsin(n)x

\int_{- π}^{π} sin (m) x sin (n) x d x

\frac{2 π ^{3}}{3} sin (m) sin (n)

Solution steps

\int_{- π}^{π} sin (m) x sin (n) x d x

Take the constant out:

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

= sin (m) sin (n) \cdot \int_{- π}^{π} xx d x

Simplify

xx : x^{2}

= sin (m) sin (n) \cdot \int_{- π}^{π} x^{2} d x

Apply the Power Rule

: [\frac{x ^{3}}{3}]_{- π}^{π}

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

Compute the boundaries:

2 \cdot \frac{π ^{3}}{3}

= sin (m) sin (n) \cdot 2 \cdot \frac{π ^{3}}{3}

Simplify

= \frac{2 π ^{3}}{3} sin (m) sin (n)

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What is the integral from-pi to pi of sin(m)xsin(n)x ?
The integral from-pi to pi of sin(m)xsin(n)x is (2pi^3)/3 sin(m)sin(n)