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dx/(tg(x)*cos2(x))
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dx divide by (tg(x)*cos^2(x))

Integral of dx/(tg(x)*cos^2(x)) dx

The solution

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  1                    
  /                    
 |                     
 |          1          
 |  1*-------------- dx
 |              2      
 |    tan(x)*cos (x)   
 |                     
/                      
0

$$\int\limits_{0}^{1} 1 \cdot \frac{1}{\cos^{2}{\left(x \right)} \tan{\left(x \right)}}\, dx$$

Integral(1/(tan(x)*cos(x)^2), (x, 0, 1))

Detail solution

Rewrite the integrand:

$\frac{\sec^{2}{\left(x \right)}}{\tan{\left(x \right)}} = \frac{\tan{\left(x \right)} \sec^{2}{\left(x \right)}}{\sec^{2}{\left(x \right)} - 1}$
There are multiple ways to do this integral.
Method #1
1. Let $u = \sec^{2}{\left(x \right)} - 1$ .
  
  Then let $du = 2 \tan{\left(x \right)} \sec^{2}{\left(x \right)} dx$ and substitute $\frac{du}{2}$ :
  
  $\int \frac{1}{4 u}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{1}{2 u}\, du = \frac{\int \frac{1}{u}\, du}{2}$
    1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
    So, the result is: $\frac{\log{\left(u \right)}}{2}$
  Now substitute $u$ back in:
  
  $\frac{\log{\left(\sec^{2}{\left(x \right)} - 1 \right)}}{2}$
Method #2
1. Let $u = \sec^{2}{\left(x \right)}$ .
  
  Then let $du = 2 \tan{\left(x \right)} \sec^{2}{\left(x \right)} dx$ and substitute $du$ :
  
  $\int \frac{1}{2 u - 2}\, du$
  1. There are multiple ways to do this integral.
    Method #1
    
    Let $u = 2 u - 2$ .
    
    Then let $du = 2 du$ and substitute $\frac{du}{2}$ :
    
    $\int \frac{1}{4 u}\, du$
    
    The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{1}{2 u}\, du = \frac{\int \frac{1}{u}\, du}{2}$
    
    The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
    
    So, the result is: $\frac{\log{\left(u \right)}}{2}$
    
    Now substitute $u$ back in:
    
    $\frac{\log{\left(2 u - 2 \right)}}{2}$
    Method #2
    
    Rewrite the integrand:
    
    $\frac{1}{2 u - 2} = \frac{1}{2 \left(u - 1\right)}$
    
    The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{1}{2 \left(u - 1\right)}\, du = \frac{\int \frac{1}{u - 1}\, du}{2}$
    
    Let $u = u - 1$ .
    
    Then let $du = du$ and substitute $du$ :
    
    $\int \frac{1}{u}\, du$
    
    The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
    
    Now substitute $u$ back in:
    
    $\log{\left(u - 1 \right)}$
    
    So, the result is: $\frac{\log{\left(u - 1 \right)}}{2}$
  Now substitute $u$ back in:
  
  $\frac{\log{\left(2 \sec^{2}{\left(x \right)} - 2 \right)}}{2}$
Method #3
1. Let $u = \sec{\left(x \right)}$ .
  
  Then let $du = \tan{\left(x \right)} \sec{\left(x \right)} dx$ and substitute $du$ :
  
  $\int \frac{u}{u^{2} - 1}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{u}{u^{2} - 1}\, du = \frac{\int \frac{2 u}{u^{2} - 1}\, du}{2}$
    1. Let $u = u^{2} - 1$ .
      
      Then let $du = 2 u du$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{1}{2 u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(u^{2} - 1 \right)}$
    So, the result is: $\frac{\log{\left(u^{2} - 1 \right)}}{2}$
  Now substitute $u$ back in:
  
  $\frac{\log{\left(\sec^{2}{\left(x \right)} - 1 \right)}}{2}$
Now simplify:

$\frac{\log{\left(\tan^{2}{\left(x \right)} \right)}}{2}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

$$\log \sin x-{{\log \left(\sin ^2x-1\right)}\over{2}}$$

Simplify

The answer [src]

     pi*I
oo - ----
      2

$${\it \%a}$$

     pi*I
oo - ----
      2

$$\infty - \frac{i \pi}{2}$$

Simplify

Numerical answer [src]

44.5334688581098

44.5334688581098

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

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Identical expressions

Integral of dx/(tg(x)*cos^2(x)) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #1

Method #2

Method #3