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

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

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

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

Solution steps

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

Expand

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

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

\int \frac{x}{cos ( x ) + 1} d x = x tan (\frac{x}{2}) + 2 ln cos (\frac{x}{2})

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

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

Add a constant to the solution

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

\int e^{2 x} (e^{2 x} + 1)^{4} d x

\int 2^{9 - x} d x

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

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

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