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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of (cos(y))/(sqrt(4+3sin(y)))

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

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

Solution steps

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

Apply u-substitution

= \int \frac{1}{3 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{1}{u} d u

Apply the Power Rule

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

Substitute back

u = 4 + 3 sin (y)

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

Simplify

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

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

Add a constant to the solution

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

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