What is the integral of cos(x)cos(3sin(x)) ?

The integral of cos(x)cos(3sin(x)) is 1/3 sin(3sin(x))+C

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integral of cos(x)cos(3sin(x))

\int cos (x) cos (3 sin (x)) d x

\frac{1}{3} sin (3 sin (x)) + C

Solution steps

\int cos (x) cos (3 sin (x)) d x

Apply u-substitution

: \int cos (3 u) d u

= \int cos (3 u) d u

Apply u-substitution

: \int cos (v) \frac{1}{3} d v

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

Take the constant out:

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

= \frac{1}{3} \cdot \int cos (v) d v

Use the common integral:

\int cos (v) d v = sin (v)

= \frac{1}{3} sin (v)

Substitute back

= \frac{1}{3} sin (3 sin (x))

Add a constant to the solution

= \frac{1}{3} sin (3 sin (x)) + C

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What is the integral of cos(x)cos(3sin(x)) ?
The integral of cos(x)cos(3sin(x)) is 1/3 sin(3sin(x))+C