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

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

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

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

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

Solution steps

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

= - \frac{1}{2} ln ∣ 1 + cos (2 x) ∣

Simplify

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

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

Add a constant to the solution

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

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