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Solving integrals/ sqrt(12-x^2)

Integral of sqrt(12-x^2) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
     ___               
 2*\/ 3                
    /                  
   |                   
   |       _________   
   |      /       2    
   |    \/  12 - x   dx
   |                   
  /                    
  0
$$\int\limits_{0}^{2 \sqrt{3}} \sqrt{12 - x^{2}}\, dx$$ 
Integral(sqrt(12 - x^2), (x, 0, 2*sqrt(3)))
Detail solution

    TrigSubstitutionRule(theta=_theta, func=2*sqrt(3)*sin(_theta), rewritten=12*cos(_theta)**2, substep=ConstantTimesRule(constant=12, other=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), context=12*cos(_theta)**2, symbol=_theta), restriction=(x > -2*sqrt(3)) & (x < 2*sqrt(3)), context=sqrt(12 - x**2), symbol=x)

  1. Add the constant of integration:

The answer is:

The answer (Indefinite) [src] 
  /                                                                                             
 |                                                                                              
 |    _________          //                       _________                                    \
 |   /       2           ||      /    ___\       /       2                                     |
 | \/  12 - x   dx = C + |<      |x*\/ 3 |   x*\/  12 - x           /         ___          ___\|
 |                       ||6*asin|-------| + --------------  for And\x > -2*\/ 3 , x < 2*\/ 3 /|
/                        \\      \   6   /         2                                           /
$$\int \sqrt{12 - x^{2}}\, dx = C + \begin{cases} \frac{x \sqrt{12 - x^{2}}}{2} + 6 \operatorname{asin}{\left(\frac{\sqrt{3} x}{6} \right)} & \text{for}\: x > - 2 \sqrt{3} \wedge x < 2 \sqrt{3} \end{cases}$$
Simplify
The graph
Plot the graph f(x)
The answer [src] 
3*pi
$$3 \pi$$ 
=
= 
3*pi
$$3 \pi$$ 
3*pi
Numerical answer [src] 
9.42477796076938
9.42477796076938

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

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