What is the integral from-pi to pi of cos(2x)cos(nx) ?

The integral from-pi to pi of cos(2x)cos(nx) is 0

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integral from-pi to pi of cos(2x)cos(nx)

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

0

Solution steps

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

Rewrite using trig identities

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

\int_{- π}^{π} cos (2 x + n x) d x = 0

\int_{- π}^{π} cos (2 x - n x) d x = 0

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

Simplify

= 0

\int \frac{x}{x ^{2} - 5} d x

7^{x^{4} + x}

\int (sin (t) 1 + cos (t)) d t

y = 3 x - 2 x, (1, 1)

(2 e^{y} - x) \frac{d y}{d x} = 1

What is the integral from-pi to pi of cos(2x)cos(nx) ?
The integral from-pi to pi of cos(2x)cos(nx) is 0