Integral of (1+sinx)/(1+cosx) dx

The solution

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

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

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

Detail solution

Rewrite the integrand:

$\frac{\sin{\left(x \right)} + 1}{\cos{\left(x \right)} + 1} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)} + 1} + \frac{1}{\cos{\left(x \right)} + 1}$
Integrate term-by-term:
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:
  
  $- \log{\left(\cos{\left(x \right)} + 1 \right)}$
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $\tan{\left(\frac{x}{2} \right)}$
The result is: $- \log{\left(\cos{\left(x \right)} + 1 \right)} + \tan{\left(\frac{x}{2} \right)}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

   /       2     \           
log\1 + tan (1/2)/ + tan(1/2)

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

   /       2     \           
log\1 + tan (1/2)/ + tan(1/2)

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

log(1 + tan(1/2)^2) + tan(1/2)

Numerical answer [src]

0.807470970731236

0.807470970731236

The graph

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

Other calculators

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Identical expressions

Similar expressions

Integral of (1+sinx)/(1+cosx) dx

Limits of integration:

The graph:

Piecewise:

The solution