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

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

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

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

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

Solution steps

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

Rewrite using trig identities

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

= \frac{1}{2} ln ∣ - cos (2 x) + 2 ∣

Add a constant to the solution

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

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