What is the integral of cos(y)cos(4y) ?

The integral of cos(y)cos(4y) is 1/2 (1/3 sin(3y)+1/5 sin(5y))+C

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integral of cos(y)cos(4y)

\int cos (y) cos (4 y) d y

\frac{1}{2} (\frac{1}{3} sin (3 y) + \frac{1}{5} sin (5 y)) + C

Solution steps

\int cos (y) cos (4 y) d y

Rewrite using trig identities

= \int \frac{1}{2} (cos (5 y) + cos (- 3 y)) d y

Take the constant out:

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

= \frac{1}{2} \cdot \int cos (5 y) + cos (- 3 y) d y

Simplify

cos (5 y) + cos (- 3 y) : cos (3 y) + cos (5 y)

= \frac{1}{2} \cdot \int cos (3 y) + cos (5 y) d y

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 (3 y) d y + \int cos (5 y) d y)

\int cos (3 y) d y = \frac{1}{3} sin (3 y)

\int cos (5 y) d y = \frac{1}{5} sin (5 y)

= \frac{1}{2} (\frac{1}{3} sin (3 y) + \frac{1}{5} sin (5 y))

Add a constant to the solution

= \frac{1}{2} (\frac{1}{3} sin (3 y) + \frac{1}{5} sin (5 y)) + C

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What is the integral of cos(y)cos(4y) ?
The integral of cos(y)cos(4y) is 1/2 (1/3 sin(3y)+1/5 sin(5y))+C