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

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  • 1/sqrt(1-(x/2)^2)dx

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

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

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1                   
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 |         1          
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0
∫0111−(x2)2 dx\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

  1. Rewrite the integrand:

    11−(x2)2=24−x2\frac{1}{\sqrt{1 - \left(\frac{x}{2}\right)^{2}}} = \frac{2}{\sqrt{4 - x^{2}}}

  2. The integral of a constant times a function is the constant times the integral of the function:

    ∫24−x2 dx=2∫14−x2 dx\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:

      ∫14−x2 dx=∫11−x24 dx2\int \frac{1}{\sqrt{4 - x^{2}}}\, dx = \frac{\int \frac{1}{\sqrt{1 - \frac{x^{2}}{4}}}\, dx}{2}

      1. Let u=x2u = \frac{x}{2}.

        Then let du=dx2du = \frac{dx}{2} and substitute 2du2 du:

        ∫41−u2 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:

          ∫21−u2 du=2∫11−u2 du\int \frac{2}{\sqrt{1 - u^{2}}}\, du = 2 \int \frac{1}{\sqrt{1 - u^{2}}}\, du

            ArcsinRule(context=1/sqrt(1 - _u**2), symbol=_u)

          So, the result is: 2asin⁡(u)2 \operatorname{asin}{\left(u \right)}

        Now substitute uu back in:

        2asin⁡(x2)2 \operatorname{asin}{\left(\frac{x}{2} \right)}

      So, the result is: asin⁡(x2)\operatorname{asin}{\left(\frac{x}{2} \right)}

    So, the result is: 2asin⁡(x2)2 \operatorname{asin}{\left(\frac{x}{2} \right)}

  3. Add the constant of integration:

    2asin⁡(x2)+constant2 \operatorname{asin}{\left(\frac{x}{2} \right)}+ \mathrm{constant}

The answer is:

2asin⁡(x2)+constant2 \operatorname{asin}{\left(\frac{x}{2} \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                  
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 |        1                       /x\
 | --------------- dx = C + 2*asin|-|
 |      __________                \2/
 |     /        2                    
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/
∫11−(x2)2 dx=C+2asin⁡(x2)\int \frac{1}{\sqrt{1 - \left(\frac{x}{2}\right)^{2}}}\, dx = C + 2 \operatorname{asin}{\left(\frac{x}{2} \right)}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.9002
Plot the graph f(x)
The answer [src] 
  1                               
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 |  |     -I             x        
 |  |--------------  for -- > 1   
 |  |     _________      4        
 |  |    /       2                
 |  |   /       x                 
 |  |  /   -1 + --                
 |  |\/         4                 
 |  <                           dx
 |  |      1                      
 |  |-------------   otherwise    
 |  |     ________                
 |  |    /      2                 
 |  |   /      x                  
 |  |  /   1 - --                 
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0
∫01{−ix24−1for x24>111−x24otherwise dx\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                               
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 |  |     -I             x        
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 |  |     _________      4        
 |  |    /       2                
 |  |   /       x                 
 |  |  /   -1 + --                
 |  |\/         4                 
 |  <                           dx
 |  |      1                      
 |  |-------------   otherwise    
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0
∫01{−ix24−1for x24>111−x24otherwise dx\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
Integral of 1/sqrt(1-(x/2)^2) dx

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

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