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

The integral of (sin(2x))/(sqrt(1+cos(2x))) is -sqrt(2)cos(x)+C

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

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

- 2 cos (x) + 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

Apply the Power Rule

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

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

Substitute back

u = 1 + cos (2 x)

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

Simplify

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

= - 2 cos (x)

Add a constant to the solution

= - 2 cos (x) + C

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