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

The solution

You have entered [src]

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

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

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

Detail solution

Rewrite the integrand:

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

oo

$$\infty$$

oo

$$\infty$$

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Numerical answer [src]

88.096853256335

88.096853256335

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

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

Limits of integration:

The graph:

Piecewise:

The solution