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

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

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integral of (cos(3x))/(sqrt(sin(3x)))

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

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

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

= \frac{1}{3} \cdot \int \frac{cos ( u )}{sin ( u )} d u

Apply u-substitution

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

= \frac{1}{3} \cdot \int \frac{1}{v} d v

Apply the Power Rule

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

= \frac{1}{3} \cdot 2 v^{\frac{1}{2}}

Substitute back

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

Simplify

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

= \frac{2}{3} sin (3 x)

Add a constant to the solution

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

f (x) = - 6 x

\frac{d}{d x} (6 (x^{2} + 4)^{8})

\frac{d}{d x} (\frac{π x}{3})

f (x) = x^{2} - 7 x + 7, (0, 7)

x \to 1 + lim (4 x - 2 x^{2})

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