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Solving integrals/ 1/sqrt(1-(x/2)^2)

Integral of 1/sqrt(1-(x/2)^2) dx

The solution

You have entered [src]

  1                   
  /                   
 |                    
 |         1          
 |  --------------- dx
 |       __________   
 |      /        2    
 |     /      /x\     
 |    /   1 - |-|     
 |  \/        \2/     
 |                    
/                     
0

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

Integral(1/(sqrt(1 - (x/2)^2)), (x, 0, 1))

Detail solution

Rewrite the integrand:

$\frac{1}{\sqrt{1 - \left(\frac{x}{2}\right)^{2}}} = \frac{2}{\sqrt{4 - x^{2}}}$
The integral of a constant times a function is the constant times the integral of the function:

$\int \frac{2}{\sqrt{4 - x^{2}}}\, dx = 2 \int \frac{1}{\sqrt{4 - x^{2}}}\, dx$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{1}{\sqrt{4 - x^{2}}}\, dx = \frac{\int \frac{1}{\sqrt{1 - \frac{x^{2}}{4}}}\, dx}{2}$
  1. Let $u = \frac{x}{2}$ .
    
    Then let $du = \frac{dx}{2}$ and substitute $2 du$ :
    
    $\int \frac{4}{\sqrt{1 - u^{2}}}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{2}{\sqrt{1 - u^{2}}}\, du = 2 \int \frac{1}{\sqrt{1 - u^{2}}}\, du$
      So, the result is: $2 \operatorname{asin}{\left(u \right)}$
    Now substitute $u$ back in:
    
    $2 \operatorname{asin}{\left(\frac{x}{2} \right)}$
  So, the result is: $\operatorname{asin}{\left(\frac{x}{2} \right)}$
So, the result is: $2 \operatorname{asin}{\left(\frac{x}{2} \right)}$
Add the constant of integration:

$2 \operatorname{asin}{\left(\frac{x}{2} \right)}+ \mathrm{constant}$

The answer is:

$2 \operatorname{asin}{\left(\frac{x}{2} \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                  
 |                                   
 |        1                       /x\
 | --------------- dx = C + 2*asin|-|
 |      __________                \2/
 |     /        2                    
 |    /      /x\                     
 |   /   1 - |-|                     
 | \/        \2/                     
 |                                   
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

  1                               
  /                               
 |                                
 |  /                     2       
 |  |     -I             x        
 |  |--------------  for -- > 1   
 |  |     _________      4        
 |  |    /       2                
 |  |   /       x                 
 |  |  /   -1 + --                
 |  |\/         4                 
 |  <                           dx
 |  |      1                      
 |  |-------------   otherwise    
 |  |     ________                
 |  |    /      2                 
 |  |   /      x                  
 |  |  /   1 - --                 
 |  |\/        4                  
 |  \                             
 |                                
/                                 
0

$$\int\limits_{0}^{1} \begin{cases} - \frac{i}{\sqrt{\frac{x^{2}}{4} - 1}} & \text{for}\: \frac{x^{2}}{4} > 1 \\\frac{1}{\sqrt{1 - \frac{x^{2}}{4}}} & \text{otherwise} \end{cases}\, dx$$

  1                               
  /                               
 |                                
 |  /                     2       
 |  |     -I             x        
 |  |--------------  for -- > 1   
 |  |     _________      4        
 |  |    /       2                
 |  |   /       x                 
 |  |  /   -1 + --                
 |  |\/         4                 
 |  <                           dx
 |  |      1                      
 |  |-------------   otherwise    
 |  |     ________                
 |  |    /      2                 
 |  |   /      x                  
 |  |  /   1 - --                 
 |  |\/        4                  
 |  \                             
 |                                
/                                 
0

$$\int\limits_{0}^{1} \begin{cases} - \frac{i}{\sqrt{\frac{x^{2}}{4} - 1}} & \text{for}\: \frac{x^{2}}{4} > 1 \\\frac{1}{\sqrt{1 - \frac{x^{2}}{4}}} & \text{otherwise} \end{cases}\, dx$$

Integral(Piecewise((-i/sqrt(-1 + x^2/4), x^2/4 > 1), (1/sqrt(1 - x^2/4), True)), (x, 0, 1))

Numerical answer [src]

1.0471975511966

1.0471975511966

The graph

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

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How to use it?

Identical expressions

Similar expressions

Integral of 1/sqrt(1-(x/2)^2) dx

Limits of integration:

The graph:

Piecewise:

The solution