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

The integral of (sin(x))/(2+5cos(x)) is -1/5 ln|2+5cos(x)|+C

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

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

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

Solution steps

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

Apply u-substitution

: \int - \frac{1}{5 u} d u

= \int - \frac{1}{5 u} d u

Take the constant out:

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

= - \frac{1}{5} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - \frac{1}{5} ln ∣ u ∣

Substitute back

u = 2 + 5 cos (x)

= - \frac{1}{5} ln ∣ 2 + 5 cos (x) ∣

Add a constant to the solution

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

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What is the integral of (sin(x))/(2+5cos(x)) ?
The integral of (sin(x))/(2+5cos(x)) is -1/5 ln|2+5cos(x)|+C