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Solving integrals/ cos2x/(1+sin2x)

Integral of cos2x/(1+sin2x) dx

The solution

You have entered [src]

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

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

Integral(cos(2*x)/(1 + 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 $du$ :
  
  $\int \frac{\cos{\left(u \right)}}{2 \sin{\left(u \right)} + 2}\, du$
  1. Let $u = 2 \sin{\left(u \right)} + 2$ .
    
    Then let $du = 2 \cos{\left(u \right)} du$ 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:
      
      $\int \frac{1}{2 u}\, du = \frac{\int \frac{1}{u}\, du}{2}$
      
      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:
    
    $\frac{\log{\left(2 \sin{\left(u \right)} + 2 \right)}}{2}$
  Now substitute $u$ back in:
  
  $\frac{\log{\left(2 \sin{\left(2 x \right)} + 2 \right)}}{2}$
Method #2
1. Let $u = \sin{\left(2 x \right)} + 1$ .
  
  Then let $du = 2 \cos{\left(2 x \right)} dx$ 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:
  
  $\frac{\log{\left(\sin{\left(2 x \right)} + 1 \right)}}{2}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

log(1 + sin(2))
---------------
       2

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

log(1 + sin(2))
---------------
       2

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

Упростить

Numerical answer [src]

0.323367667515383

0.323367667515383

The graph

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

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Identical expressions

Similar expressions

Integral of cos2x/(1+sin2x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2