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

The integral of (sin(x))/(1+sin^2(x/2)) is 2ln|-cos(x)+3|+C

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

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

2 ln ∣ - cos (x) + 3 ∣ + C

Solution steps

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

Rewrite using trig identities

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

Apply u-substitution

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 ln ∣ u ∣

Substitute back

u = - cos (x) + 3

= 2 ln ∣ - cos (x) + 3 ∣

Add a constant to the solution

= 2 ln ∣ - cos (x) + 3 ∣ + C

What is the integral of (sin(x))/(1+sin^2(x/2)) ?
The integral of (sin(x))/(1+sin^2(x/2)) is 2ln|-cos(x)+3|+C