What is the integral of (sin(2x))/(sqrt(4-sin^2(x))) ?

The integral of (sin(2x))/(sqrt(4-sin^2(x))) is -2sqrt(cos^2(x)+3)+C

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integral of (sin(2x))/(sqrt(4-sin^2(x)))

\int \frac{sin ( 2 x )}{4 - sin ^{2} ( x )} d x

- 2 cos^{2} (x) + 3 + C

Solution steps

\int \frac{sin ( 2 x )}{4 - 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}{u} d u

= \int - \frac{1}{u} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int \frac{1}{u} d u

Apply the Power Rule

: 2 u^{\frac{1}{2}}

= - 2 u^{\frac{1}{2}}

Substitute back

u = cos^{2} (x) + 3

= - 2 (cos^{2} (x) + 3)^{\frac{1}{2}}

Apply radical rule:

a^{\frac{1}{n}} = n a

= - 2 cos^{2} (x) + 3

Add a constant to the solution

= - 2 cos^{2} (x) + 3 + C

\int \frac{22 x}{1 + x ^{2}} d x

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

\int x - (- d) x d x

\int \frac{2}{8 - x ^{2}} d x

\int \frac{x}{( x - 1 ) ( 2 - x )} d x

What is the integral of (sin(2x))/(sqrt(4-sin^2(x))) ?
The integral of (sin(2x))/(sqrt(4-sin^2(x))) is -2sqrt(cos^2(x)+3)+C