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Solving integrals/ sqrt(-(x)^2+5)

Integral of sqrt(-(x)^2+5) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  2                 
  /                 
 |                  
 |     __________   
 |    /    2        
 |  \/  - x  + 5  dx
 |                  
/                   
0
$$\int\limits_{0}^{2} \sqrt{5 - x^{2}}\, dx$$ 
Integral(sqrt(-x^2 + 5), (x, 0, 2))
Detail solution

    TrigSubstitutionRule(theta=_theta, func=sqrt(5)*sin(_theta), rewritten=5*cos(_theta)**2, substep=ConstantTimesRule(constant=5, 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=5*cos(_theta)**2, symbol=_theta), restriction=(x < sqrt(5)) & (x > -sqrt(5)), context=sqrt(5 - x**2), symbol=x)

  1. Add the constant of integration:

The answer is:

The answer (Indefinite) [src] 
  /                                                                                         
 |                        //      /    ___\                                                \
 |    __________          ||      |x*\/ 5 |        ________                                |
 |   /    2               ||5*asin|-------|       /      2                                 |
 | \/  - x  + 5  dx = C + |<      \   5   /   x*\/  5 - x           /       ___        ___\|
 |                        ||--------------- + -------------  for And\x > -\/ 5 , x < \/ 5 /|
/                         ||       2                2                                      |
                          \\                                                               /
$$\int \sqrt{5 - x^{2}}\, dx = C + \begin{cases} \frac{x \sqrt{5 - x^{2}}}{2} + \frac{5 \operatorname{asin}{\left(\frac{\sqrt{5} x}{5} \right)}}{2} & \text{for}\: x > - \sqrt{5} \wedge x < \sqrt{5} \end{cases}$$
Simplify
The graph
Plot the graph f(x)
The answer [src] 
          /    ___\
          |2*\/ 5 |
    5*asin|-------|
          \   5   /
1 + ---------------
           2
$$1 + \frac{5 \operatorname{asin}{\left(\frac{2 \sqrt{5}}{5} \right)}}{2}$$ 
=
= 
          /    ___\
          |2*\/ 5 |
    5*asin|-------|
          \   5   /
1 + ---------------
           2
$$1 + \frac{5 \operatorname{asin}{\left(\frac{2 \sqrt{5}}{5} \right)}}{2}$$ 
1 + 5*asin(2*sqrt(5)/5)/2
Numerical answer [src] 
3.76787179448523
3.76787179448523

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

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