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Solving integrals/ cos(2*x)/sin(2*x)

Integral of cos(2x)/sin(2x) dx

The solution

You have entered [src]

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

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

Integral(cos(2*x)/sin(2*x), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = 2 x$ .
  
  Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
  
  $\int \frac{\cos{\left(u \right)}}{2 \sin{\left(u \right)}}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\cos{\left(u \right)}}{\sin{\left(u \right)}}\, du = \frac{\int \frac{\cos{\left(u \right)}}{\sin{\left(u \right)}}\, du}{2}$
    1. Let $u = \sin{\left(u \right)}$ .
      
      Then let $du = \cos{\left(u \right)} du$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(\sin{\left(u \right)} \right)}$
    So, the result is: $\frac{\log{\left(\sin{\left(u \right)} \right)}}{2}$
  Now substitute $u$ back in:
  
  $\frac{\log{\left(\sin{\left(2 x \right)} \right)}}{2}$
Method #2
1. Let $u = \sin{\left(2 x \right)}$ .
  
  Then let $du = 2 \cos{\left(2 x \right)} dx$ and substitute $\frac{du}{2}$ :
  
  $\int \frac{1}{2 u}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{1}{u}\, du = \frac{\int \frac{1}{u}\, du}{2}$
    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:
  
  $\frac{\log{\left(\sin{\left(2 x \right)} \right)}}{2}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

oo

$$\infty$$

oo

$$\infty$$

oo

Numerical answer [src]

21.6511079586689

21.6511079586689

The graph

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of cos(2x)/sin(2x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of cos(2*x)/sin(2*x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Integral of cos(2x)/sin(2x) dx