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

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

\int_{0}^{π} sin (x) 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 (4 x - 3)^{6} d x

t an g e n t f (x) = \frac{6}{x}, at x = 1

x \to 4 - lim (\frac{x - 4}{x - 1})

d er i v a t i v e y = \frac{1}{x}

\int 7 x^{2} - 3 x + 8 + \frac{1}{x} + \frac{2}{x ^{2}} d x

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