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

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

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

\int_{0}^{π} sin (x) \cdot sin (n x) d x

0

Solution steps

\int_{0}^{π} sin (x) sin (n x) d x

Rewrite using trig identities

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

\int_{0}^{π} cos (x + n x) d x = 0

\int_{0}^{π} cos (x - n x) d x = 0

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

Simplify

= 0

\int_{0}^{1} - x + 1 d x

\int_{0}^{1} 4 x f (x^{2} + 1) d x

\int_{0}^{\frac{π}{2}} \frac{sin ( x )}{x} d x

\int_{0}^{\frac{5}{6}} \frac{1}{36 x ^{2} + 25} d x

\int_{0}^{\frac{π}{2}} cos^{3} (θ) d θ

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