What is the integral of (cos(3x))/(sqrt(1-2sin(3x))) ?

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of (cos(3x))/(sqrt(1-2sin(3x)))

\int \frac{cos ( 3 x )}{1 - 2 sin ( 3 x )} d x

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

Solution steps

\int \frac{cos ( 3 x )}{1 - 2 sin ( 3 x )} d x

Apply u-substitution

= \int - \frac{1}{6 u} d u

Take the constant out:

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

= - \frac{1}{6} \cdot \int \frac{1}{u} d u

Apply the Power Rule

= - \frac{1}{6} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = 1 - 2 sin (3 x)

= - \frac{1}{6} \cdot 2 (1 - 2 sin (3 x))^{\frac{1}{2}}

Simplify

- \frac{1}{6} \cdot 2 (1 - 2 sin (3 x))^{\frac{1}{2}} : - \frac{1}{3} 1 - 2 sin (3 x)

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

Add a constant to the solution

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

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