What is the integral of 1/(4+4sin(x)) ?

The integral of 1/(4+4sin(x)) is -1/(2(tan(x/2)+1))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of 1/(4+4sin(x))

\int \frac{1}{4 + 4 sin ( x )} d x

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

Solution steps

\int \frac{1}{4 + 4 sin ( x )} d x

Apply u-substitution

= \int \frac{1}{2 ( u ^{2} + 2 u + 1 )} 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{1}{u ^{2} + 2 u + 1} d u

Complete the square

u^{2} + 2 u + 1 : (u + 1)^{2}

= \frac{1}{2} \cdot \int \frac{1}{( u + 1 ) ^{2}} d u

Apply u-substitution

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

Apply the Power Rule

= \frac{1}{2} (- \frac{1}{v})

Substitute back

= \frac{1}{2} (- \frac{1}{tan ( \frac{x}{2} ) + 1})

Simplify

\frac{1}{2} (- \frac{1}{tan ( \frac{x}{2} ) + 1}) : - \frac{1}{2 ( tan ( \frac{x}{2} ) + 1 )}

= - \frac{1}{2 ( tan ( \frac{x}{2} ) + 1 )}

Add a constant to the solution

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

What is the integral of 1/(4+4sin(x)) ?
The integral of 1/(4+4sin(x)) is -1/(2(tan(x/2)+1))+C