What is the integral of (cos(2x))/(3+sin(2x)) ?

The integral of (cos(2x))/(3+sin(2x)) is 1/2 ln|3+sin(2x)|+C

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integral of (cos(2x))/(3+sin(2x))

\int \frac{cos ( 2 x )}{3 + sin ( 2 x )} d x

\frac{1}{2} ln ∣ 3 + sin (2 x) ∣ + C

Solution steps

\int \frac{cos ( 2 x )}{3 + sin ( 2 x )} d x

Apply u-substitution

: \int \frac{1}{2 u} d u

= \int \frac{1}{2 u} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{2} ln ∣ u ∣

Substitute back

u = 3 + sin (2 x)

= \frac{1}{2} ln ∣ 3 + sin (2 x) ∣

Add a constant to the solution

= \frac{1}{2} ln ∣ 3 + sin (2 x) ∣ + C

\int (5 x x - \frac{1}{x ^{3}} + 3) d x

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\int \frac{5}{6 - 4 x} d x

What is the integral of (cos(2x))/(3+sin(2x)) ?
The integral of (cos(2x))/(3+sin(2x)) is 1/2 ln|3+sin(2x)|+C