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  • Identical expressions

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  • one divide by two multiply by square root of (one minus x to the power of two)
  • 1/2*√(1-x^2)
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  • 1/2sqrt(1-x^2)
  • 1/2sqrt(1-x2)
  • 1/2sqrt1-x2
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  • 1/2*sqrt(1-x^2)dx

  • Similar expressions

  • 1/2*sqrt(1+x^2)
Solving integrals/ 1/2*sqrt(1-x^2)

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

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
   ___              
 \/ 3               
 -----              
   2                
   /                
  |                 
  |      ________   
  |     /      2    
  |   \/  1 - x     
  |   ----------- dx
  |        2        
  |                 
 /                  
  ___               
\/ 2                
-----               
  2
$$\int\limits_{\frac{\sqrt{2}}{2}}^{\frac{\sqrt{3}}{2}} \frac{\sqrt{1 - x^{2}}}{2}\, dx$$ 
Integral(sqrt(1 - x^2)/2, (x, sqrt(2)/2, sqrt(3)/2))
Detail solution

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

      TrigSubstitutionRule(theta=_theta, func=sin(_theta), rewritten=cos(_theta)**2, substep=RewriteRule(rewritten=cos(2*_theta)/2 + 1/2, substep=AddRule(substeps=[ConstantTimesRule(constant=1/2, other=cos(2*_theta), substep=URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(2*_theta), symbol=_theta), context=cos(2*_theta)/2, symbol=_theta), ConstantRule(constant=1/2, context=1/2, symbol=_theta)], context=cos(2*_theta)/2 + 1/2, symbol=_theta), context=cos(_theta)**2, symbol=_theta), restriction=(x > -1) & (x < 1), context=sqrt(1 - x**2), symbol=x)

    So, the result is:

  2. Now simplify:

  3. Add the constant of integration:

The answer is:

The answer (Indefinite) [src] 
  /                     /               ________                        
 |                      |              /      2                         
 |    ________           -1, x < 1)
 | \/  1 - x            \   2            2                              
 | ----------- dx = C + ------------------------------------------------
 |      2                                      2                        
 |                                                                      
/
$$\int \frac{\sqrt{1 - x^{2}}}{2}\, dx = C + \frac{\begin{cases} \frac{x \sqrt{1 - x^{2}}}{2} + \frac{\operatorname{asin}{\left(x \right)}}{2} & \text{for}\: x > -1 \wedge x < 1 \end{cases}}{2}$$
Simplify
The graph
Plot the graph f(x)
The answer [src] 
        ___     
  1   \/ 3    pi
- - + ----- + --
  8     16    48
$$- \frac{1}{8} + \frac{\pi}{48} + \frac{\sqrt{3}}{16}$$ 
=
= 
        ___     
  1   \/ 3    pi
- - + ----- + --
  8     16    48
$$- \frac{1}{8} + \frac{\pi}{48} + \frac{\sqrt{3}}{16}$$ 
-1/8 + sqrt(3)/16 + pi/48
Numerical answer [src] 
0.0487030224228422
0.0487030224228422

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

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