What is the integral of (2sin(x))/(5+3cos(x)) ?

The integral of (2sin(x))/(5+3cos(x)) is -2/3 ln|5+3cos(x)|+C

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integral of (2sin(x))/(5+3cos(x))

\int \frac{2 sin ( x )}{5 + 3 cos ( x )} d x

- \frac{2}{3} ln ∣ 5 + 3 cos (x) ∣ + C

Solution steps

\int \frac{2 sin ( x )}{5 + 3 cos ( x )} d x

Take the constant out:

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

= 2 \cdot \int \frac{sin ( x )}{5 + 3 cos ( x )} d x

Apply u-substitution

: \int - \frac{1}{3 u} d u

= 2 \cdot \int - \frac{1}{3 u} d u

Take the constant out:

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

= 2 (- \frac{1}{3} \cdot \int \frac{1}{u} d u)

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 (- \frac{1}{3} ln ∣ u ∣)

Substitute back

u = 5 + 3 cos (x)

= 2 (- \frac{1}{3} ln ∣ 5 + 3 cos (x) ∣)

Simplify

2 (- \frac{1}{3} ln ∣ 5 + 3 cos (x) ∣) : - \frac{2}{3} ln ∣ 5 + 3 cos (x) ∣

= - \frac{2}{3} ln ∣ 5 + 3 cos (x) ∣

Add a constant to the solution

= - \frac{2}{3} ln ∣ 5 + 3 cos (x) ∣ + C

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What is the integral of (2sin(x))/(5+3cos(x)) ?
The integral of (2sin(x))/(5+3cos(x)) is -2/3 ln|5+3cos(x)|+C