What is the integral from 0 to pi/2 of cos(5t)cos(10t) ?

The integral from 0 to pi/2 of cos(5t)cos(10t) is 1/15

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integral from 0 to pi/2 of cos(5t)cos(10t)

\int_{0}^{\frac{π}{2}} cos (5 t) cos (10 t) d t

\frac{1}{15}

Solution steps

\int_{0}^{\frac{π}{2}} cos (5 t) cos (10 t) d t

Rewrite using trig identities

= \int_{0}^{\frac{π}{2}} \frac{1}{2} (cos (15 t) + cos (- 5 t)) d t

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int_{0}^{\frac{π}{2}} cos (15 t) + cos (- 5 t) d t

Simplify

cos (15 t) + cos (- 5 t) : cos (5 t) + cos (15 t)

= \frac{1}{2} \cdot \int_{0}^{\frac{π}{2}} cos (5 t) + cos (15 t) d t

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}^{\frac{π}{2}} cos (5 t) d t + \int_{0}^{\frac{π}{2}} cos (15 t) d t)

\int_{0}^{\frac{π}{2}} cos (5 t) d t = \frac{1}{5}

\int_{0}^{\frac{π}{2}} cos (15 t) d t = - \frac{1}{15}

= \frac{1}{2} (\frac{1}{5} - \frac{1}{15})

Simplify

\frac{1}{2} (\frac{1}{5} - \frac{1}{15}) : \frac{1}{15}

= \frac{1}{15}

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What is the integral from 0 to pi/2 of cos(5t)cos(10t) ?
The integral from 0 to pi/2 of cos(5t)cos(10t) is 1/15