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Solving integrals/ sin2x/(sqrt(1+cos^2x))

Integral of sin2x/(sqrt(1+cos^2x)) dx

The solution

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

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

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

Detail solution

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

$- 2 \sqrt{\cos^{2}{\left(x \right)} + 1}+ \mathrm{constant}$

The answer is:

$- 2 \sqrt{\cos^{2}{\left(x \right)} + 1}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

       _____________          
      /        2           ___
- 2*\/  1 + cos (1)  + 2*\/ 2

$$- 2 \sqrt{\cos^{2}{\left(1 \right)} + 1} + 2 \sqrt{2}$$

       _____________          
      /        2           ___
- 2*\/  1 + cos (1)  + 2*\/ 2

$$- 2 \sqrt{\cos^{2}{\left(1 \right)} + 1} + 2 \sqrt{2}$$

-2*sqrt(1 + cos(1)^2) + 2*sqrt(2)

Numerical answer [src]

0.555168158656819

0.555168158656819

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

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Identical expressions

Similar expressions

Integral of sin2x/(sqrt(1+cos^2x)) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2