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

The integral of (x+sin(x))/(1-cos(x)) is -xcot(x/2)+2ln|sin(x/2)|+ln|1-cos(x)|+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (x+sin(x))/(1-cos(x))

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

- x cot (\frac{x}{2}) + 2 ln sin (\frac{x}{2}) + ln ∣ 1 - cos (x) ∣ + C

Solution steps

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

Expand

\frac{x + sin ( x )}{1 - cos ( x )} : \frac{x}{1 - cos ( x )} + \frac{sin ( x )}{1 - cos ( x )}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{x}{1 - cos ( x )} d x + \int \frac{sin ( x )}{1 - cos ( x )} d x

\int \frac{x}{1 - cos ( x )} d x = - x cot (\frac{x}{2}) + 2 ln sin (\frac{x}{2})

\int \frac{sin ( x )}{1 - cos ( x )} d x = ln ∣ 1 - cos (x) ∣

= - x cot (\frac{x}{2}) + 2 ln sin (\frac{x}{2}) + ln ∣ 1 - cos (x) ∣

Add a constant to the solution

= - x cot (\frac{x}{2}) + 2 ln sin (\frac{x}{2}) + ln ∣ 1 - cos (x) ∣ + C

\int y (a + b y^{2}) d y

\int (e^{- 12 x} + 1 1^{x}) d x

\int - \frac{320}{x ^{6}} d x

\int x^{2} 3 d x

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

What is the integral of (x+sin(x))/(1-cos(x)) ?
The integral of (x+sin(x))/(1-cos(x)) is -xcot(x/2)+2ln|sin(x/2)|+ln|1-cos(x)|+C