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

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

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

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

\frac{1}{2} ln ∣ - cos (2 x) + 3 ∣ + C

Solution steps

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

Rewrite using trig identities

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

= \frac{1}{2} ln ∣ - cos (2 x) + 3 ∣

Add a constant to the solution

= \frac{1}{2} ln ∣ - cos (2 x) + 3 ∣ + C

y^{^{''}} + 49 y = - 49 sec^{2} (7 x)

(9 x^{2} + y - 1) - (4 y - x) y^{^{'}} = 0

ln (\frac{1}{1 - x})

\int \frac{1}{x ^{2}} + \frac{4}{x x} + 2 d x

\int x e^{y} d y

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