What is the integral of (cos(3x))/(sqrt(5+2sin(3x))) ?

The integral of (cos(3x))/(sqrt(5+2sin(3x))) is 1/3 sqrt(5+2sin(3x))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (cos(3x))/(sqrt(5+2sin(3x)))

\int \frac{cos ( 3 x )}{5 + 2 sin ( 3 x )} d x

\frac{1}{3} 5 + 2 sin (3 x) + C

Solution steps

\int \frac{cos ( 3 x )}{5 + 2 sin ( 3 x )} d x

Apply u-substitution

: \int \frac{cos ( u )}{3 5 + 2 sin ( u )} d u

= \int \frac{cos ( u )}{3 5 + 2 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 )}{5 + 2 sin ( u )} d u

Apply u-substitution

: \int \frac{1}{2 v} d v

= \frac{1}{3} \cdot \int \frac{1}{2 v} d v

Take the constant out:

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

= \frac{1}{3} \cdot \frac{1}{2} \cdot \int \frac{1}{v} d v

Apply the Power Rule

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

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

Substitute back

= \frac{1}{3} \cdot \frac{1}{2} \cdot 2 (5 + 2 sin (3 x))^{\frac{1}{2}}

Simplify

\frac{1}{3} \cdot \frac{1}{2} \cdot 2 (5 + 2 sin (3 x))^{\frac{1}{2}} : \frac{1}{3} 5 + 2 sin (3 x)

= \frac{1}{3} 5 + 2 sin (3 x)

Add a constant to the solution

= \frac{1}{3} 5 + 2 sin (3 x) + C

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

\int \frac{1}{( 2 x - 3 ) 5 - 12 x + 4 x ^{2}} d x

\int (t - \frac{2}{t}) (t + \frac{2}{t}) d t

\int \frac{x}{( 9 + x ^{2} ) ^{3}} d x

\int \frac{1}{( 1 + y ) y} d y

What is the integral of (cos(3x))/(sqrt(5+2sin(3x))) ?
The integral of (cos(3x))/(sqrt(5+2sin(3x))) is 1/3 sqrt(5+2sin(3x))+C