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

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

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

\int_{- π}^{π} cos (x) cos (2 x) d x

0

Solution steps

\int_{- π}^{π} cos (x) cos (2 x) d x

Apply Integration By Parts

: [cos (2 x) sin (x) - \int - 2 sin (2 x) sin (x) d x]_{- π}^{π}

= [cos (2 x) sin (x) - \int - 2 sin (2 x) sin (x) d x]_{- π}^{π}

\int - 2 sin (2 x) sin (x) d x = - \frac{4}{3} sin^{3} (x)

= [cos (2 x) sin (x) - (- \frac{4}{3} sin^{3} (x))]_{- π}^{π}

Simplify

[cos (2 x) sin (x) - (- \frac{4}{3} sin^{3} (x))]_{- π}^{π} : [\frac{1}{3} (3 cos (2 x) sin (x) + 4 sin^{3} (x))]_{- π}^{π}

= [\frac{1}{3} (3 cos (2 x) sin (x) + 4 sin^{3} (x))]_{- π}^{π}

Compute the boundaries:

0

= 0

\int_{0}^{\frac{π}{2}} \frac{1}{2} - sin (x) d x

\int_{- 2}^{3} \frac{1}{x} d x

\int_{2}^{11} x - 2 d x

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

\int_{0}^{1} x^{3} + \frac{3}{4} d x

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