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

The integral of (cos(2x))/(sqrt(3+4sin(2x))) is 1/4 sqrt(3+4sin(2x))+C

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integral of (cos(2x))/(sqrt(3+4sin(2x)))

\int \frac{cos ( 2 x )}{3 + 4 sin ( 2 x )} d x

\frac{1}{4} 3 + 4 sin (2 x) + C

Solution steps

\int \frac{cos ( 2 x )}{3 + 4 sin ( 2 x )} d x

Apply u-substitution

: \int \frac{cos ( u )}{2 3 + 4 sin ( u )} d u

= \int \frac{cos ( u )}{2 3 + 4 sin ( 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{cos ( u )}{3 + 4 sin ( u )} d u

Apply u-substitution

: \int \frac{1}{4 v} d v

= \frac{1}{2} \cdot \int \frac{1}{4 v} d v

Take the constant out:

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

= \frac{1}{2} \cdot \frac{1}{4} \cdot \int \frac{1}{v} d v

Apply the Power Rule

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

= \frac{1}{2} \cdot \frac{1}{4} \cdot 2 v^{\frac{1}{2}}

Substitute back

= \frac{1}{2} \cdot \frac{1}{4} \cdot 2 (3 + 4 sin (2 x))^{\frac{1}{2}}

Simplify

\frac{1}{2} \cdot \frac{1}{4} \cdot 2 (3 + 4 sin (2 x))^{\frac{1}{2}} : \frac{1}{4} 3 + 4 sin (2 x)

= \frac{1}{4} 3 + 4 sin (2 x)

Add a constant to the solution

= \frac{1}{4} 3 + 4 sin (2 x) + C

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What is the integral of (cos(2x))/(sqrt(3+4sin(2x))) ?
The integral of (cos(2x))/(sqrt(3+4sin(2x))) is 1/4 sqrt(3+4sin(2x))+C