What is the integral of (sin(x))/(2cos(x)+1) ?

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

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integral of (sin(x))/(2cos(x)+1)

\int \frac{sin ( x )}{2 cos ( x ) + 1} d x

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

Solution steps

\int \frac{sin ( x )}{2 cos ( x ) + 1} 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 = 2 cos (x) + 1

= - \frac{1}{2} ln ∣ 2 cos (x) + 1 ∣

Add a constant to the solution

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

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