What is the integral of e^{2ln(sqrt(cos(3x)))} ?

The integral of e^{2ln(sqrt(cos(3x)))} is 1/3 sin(3x)+C

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integral of e^{2ln(sqrt(cos(3x)))}

\int e^{2 l n (c o s (3 x))} d x

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

Solution steps

\int e^{2 l n (c o s (3 x))} d x

Simplify

e^{2 l n (c o s (3 x))} : cos (3 x)

= \int cos (3 x) d x

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

u = 3 x

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

Add a constant to the solution

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

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What is the integral of e^{2ln(sqrt(cos(3x)))} ?
The integral of e^{2ln(sqrt(cos(3x)))} is 1/3 sin(3x)+C