What is the integral of cos(7t)cos(3t) ?

The integral of cos(7t)cos(3t) is 1/2 (1/10 sin(10t)+1/4 sin(4t))+C

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integral of cos(7t)cos(3t)

\int cos (7 t) cos (3 t) d t

\frac{1}{2} (\frac{1}{10} sin (10 t) + \frac{1}{4} sin (4 t)) + C

Solution steps

\int cos (7 t) cos (3 t) d t

Rewrite using trig identities

= \int \frac{1}{2} (cos (10 t) + cos (4 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 cos (10 t) + cos (4 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 cos (10 t) d t + \int cos (4 t) d t)

\int cos (10 t) d t = \frac{1}{10} sin (10 t)

\int cos (4 t) d t = \frac{1}{4} sin (4 t)

= \frac{1}{2} (\frac{1}{10} sin (10 t) + \frac{1}{4} sin (4 t))

Add a constant to the solution

= \frac{1}{2} (\frac{1}{10} sin (10 t) + \frac{1}{4} sin (4 t)) + C

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What is the integral of cos(7t)cos(3t) ?
The integral of cos(7t)cos(3t) is 1/2 (1/10 sin(10t)+1/4 sin(4t))+C