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

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

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integral from 0 to pi of sin(nx)sin(x)

\int_{0}^{π} sin (n x) sin (x) d x

0

Solution steps

\int_{0}^{π} sin (n x) sin (x) d x

Rewrite using trig identities

= \int_{0}^{π} \frac{- cos ( n x + x ) + cos ( n x - 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_{0}^{π} - cos (n x + x) + cos (n x - 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_{0}^{π} cos (n x + x) d x + \int_{0}^{π} cos (n x - x) d x)

\int_{0}^{π} cos (n x + x) d x = 0

\int_{0}^{π} cos (n x - x) d x = 0

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

Simplify

= 0

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