What is the integral of (cos(4x))/(2+sin(4x)) ?

The integral of (cos(4x))/(2+sin(4x)) is 1/4 ln|2+sin(4x)|+C

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integral of (cos(4x))/(2+sin(4x))

\int \frac{cos ( 4 x )}{2 + sin ( 4 x )} d x

\frac{1}{4} ln ∣ 2 + sin (4 x) ∣ + C

Solution steps

\int \frac{cos ( 4 x )}{2 + sin ( 4 x )} d x

Apply u-substitution

: \int \frac{1}{4 u} d u

= \int \frac{1}{4 u} d u

Take the constant out:

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

= \frac{1}{4} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{4} ln ∣ u ∣

Substitute back

u = 2 + sin (4 x)

= \frac{1}{4} ln ∣ 2 + sin (4 x) ∣

Add a constant to the solution

= \frac{1}{4} ln ∣ 2 + sin (4 x) ∣ + C

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What is the integral of (cos(4x))/(2+sin(4x)) ?
The integral of (cos(4x))/(2+sin(4x)) is 1/4 ln|2+sin(4x)|+C