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(x+sin(x))/(1+cos(x))dx
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Solving integrals/ (x+sin(x))/(1+cos(x))

Integral of (x+sin(x))/(1+cos(x)) dx

The solution

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  1              
  /              
 |               
 |  x + sin(x)   
 |  ---------- dx
 |  1 + cos(x)   
 |               
/                
0

$$\int\limits_{0}^{1} \frac{x + \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}\, dx$$

Integral((x + sin(x))/(1 + cos(x)), (x, 0, 1))

Detail solution

Rewrite the integrand:

$\frac{x + \sin{\left(x \right)}}{\cos{\left(x \right)} + 1} = \frac{x}{\cos{\left(x \right)} + 1} + \frac{\sin{\left(x \right)}}{\cos{\left(x \right)} + 1}$
Integrate term-by-term:
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $x \tan{\left(\frac{x}{2} \right)} - \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}$
1. Let $u = \cos{\left(x \right)} + 1$ .
  
  Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$ :
  
  $\int \left(- \frac{1}{u}\right)\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{1}{u}\, du = - \int \frac{1}{u}\, du$
    1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
    So, the result is: $- \log{\left(u \right)}$
  Now substitute $u$ back in:
  
  $- \log{\left(\cos{\left(x \right)} + 1 \right)}$
The result is: $x \tan{\left(\frac{x}{2} \right)} - \log{\left(\cos{\left(x \right)} + 1 \right)} - \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}$
Now simplify:

$x \tan{\left(\frac{x}{2} \right)} - \log{\left(\frac{2}{\cos{\left(x \right)} + 1} \right)} - \log{\left(\cos{\left(x \right)} + 1 \right)}$
Add the constant of integration:

$x \tan{\left(\frac{x}{2} \right)} - \log{\left(\frac{2}{\cos{\left(x \right)} + 1} \right)} - \log{\left(\cos{\left(x \right)} + 1 \right)}+ \mathrm{constant}$

The answer is:

$x \tan{\left(\frac{x}{2} \right)} - \log{\left(\frac{2}{\cos{\left(x \right)} + 1} \right)} - \log{\left(\cos{\left(x \right)} + 1 \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                 
 |                                                                  
 | x + sin(x)             /       2/x\\                          /x\
 | ---------- dx = C - log|1 + tan |-|| - log(1 + cos(x)) + x*tan|-|
 | 1 + cos(x)             \        \2//                          \2/
 |                                                                  
/

$$\int \frac{x + \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}\, dx = C + x \tan{\left(\frac{x}{2} \right)} - \log{\left(\cos{\left(x \right)} + 1 \right)} - \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

tan(1/2)

$$\tan{\left(\frac{1}{2} \right)}$$

tan(1/2)

$$\tan{\left(\frac{1}{2} \right)}$$

tan(1/2)

Numerical answer [src]

0.54630248984379

0.54630248984379

The graph

Use the examples entering the upper and lower limits of integration.

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Identical expressions

Similar expressions

Integral of (x+sin(x))/(1+cos(x)) dx

Limits of integration:

The graph:

Piecewise:

The solution