What is the integral of (4cos(x))/(3+2sin(x)) ?

The integral of (4cos(x))/(3+2sin(x)) is 2ln|3+2sin(x)|+C

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integral of (4cos(x))/(3+2sin(x))

\int \frac{4 cos ( x )}{3 + 2 sin ( x )} d x

2 ln ∣ 3 + 2 sin (x) ∣ + C

Solution steps

\int \frac{4 cos ( x )}{3 + 2 sin ( x )} d x

Take the constant out:

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

= 4 \cdot \int \frac{cos ( x )}{3 + 2 sin ( x )} d x

Apply u-substitution

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

= 4 \cdot \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

= 4 \cdot \frac{1}{2} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 4 \cdot \frac{1}{2} ln ∣ u ∣

Substitute back

u = 3 + 2 sin (x)

= 4 \cdot \frac{1}{2} ln ∣ 3 + 2 sin (x) ∣

Simplify

4 \cdot \frac{1}{2} ln ∣ 3 + 2 sin (x) ∣ : 2 ln ∣ 3 + 2 sin (x) ∣

= 2 ln ∣ 3 + 2 sin (x) ∣

Add a constant to the solution

= 2 ln ∣ 3 + 2 sin (x) ∣ + C

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What is the integral of (4cos(x))/(3+2sin(x)) ?
The integral of (4cos(x))/(3+2sin(x)) is 2ln|3+2sin(x)|+C