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

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

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

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

0

Solution steps

\int_{- π}^{π} sin (x) sin (n x) d x

Rewrite using trig identities

= \int_{- π}^{π} \frac{- cos ( x + n x ) + cos ( x - n x )}{2} d x

Take the constant out:

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

= \frac{1}{2} \cdot \int_{- π}^{π} - cos (x + n x) + cos (x - n x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (- \int_{- π}^{π} cos (x + n x) d x + \int_{- π}^{π} cos (x - n x) d x)

\int_{- π}^{π} cos (x + n x) d x = 0

\int_{- π}^{π} cos (x - n x) d x = 0

= \frac{1}{2} (- 0 + 0)

Simplify

= 0

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