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

The integral of ((x+sin(x)))/((1+cos(x))) is xtan(x/2)+2ln|cos(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 tan (\frac{x}{2}) + 2 ln cos (\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 tan (\frac{x}{2}) + 2 ln cos (\frac{x}{2})

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

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

Add a constant to the solution

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

\int \frac{x + 3}{x ^{2} ( x + 1 ) ^{2}} d x

\frac{\partial}{\partial x} (- \frac{a}{x})

y = x^{2} - 8 x, (3, - 15)

\int \frac{1}{x ^{3} + 8 x} d x

f (x) = 4 cos (2 x)

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