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Solving integrals/ sqrt(16-5x^2)

Integral of sqrt(16-5x^2) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  2                  
  /                  
 |                   
 |     ___________   
 |    /         2    
 |  \/  16 - 5*x   dx
 |                   
/                    
-2
$$\int\limits_{-2}^{2} \sqrt{16 - 5 x^{2}}\, dx$$ 
Integral(sqrt(16 - 5*x^2), (x, -2, 2))
Detail solution

    TrigSubstitutionRule(theta=_theta, func=4*sqrt(5)*sin(_theta)/5, rewritten=16*sqrt(5)*cos(_theta)**2/5, substep=ConstantTimesRule(constant=16*sqrt(5)/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=16*sqrt(5)*cos(_theta)**2/5, symbol=_theta), restriction=(x > -4*sqrt(5)/5) & (x < 4*sqrt(5)/5), context=sqrt(16 - 5*x**2), symbol=x)

  1. Now simplify:

  2. Add the constant of integration:

The answer is:

The answer (Indefinite) [src] 
  /                        //         /    /    ___\                         \                                    \
 |                         ||         |    |x*\/ 5 |              ___________|                                    |
 |    ___________          ||         |asin|-------|       ___   /         2 |                                    |
 |   /         2           ||     ___ |    \   4   /   x*\/ 5 *\/  16 - 5*x  |                                    |
 | \/  16 - 5*x   dx = C + |<16*\/ 5 *|------------- + ----------------------|         /         ___          ___\|
 |                         ||         \      2                   32          /         |    -4*\/ 5       4*\/ 5 ||
/                          ||-------------------------------------------------  for And|x > --------, x < -------||
                           ||                        5                                 \       5             5   /|
                           \\                                                                                     /
$$\int \sqrt{16 - 5 x^{2}}\, dx = C + \begin{cases} \frac{16 \sqrt{5} \left(\frac{\sqrt{5} x \sqrt{16 - 5 x^{2}}}{32} + \frac{\operatorname{asin}{\left(\frac{\sqrt{5} x}{4} \right)}}{2}\right)}{5} & \text{for}\: x > - \frac{4 \sqrt{5}}{5} \wedge x < \frac{4 \sqrt{5}}{5} \end{cases}$$
Simplify
The graph
Plot the graph f(x)
The answer [src] 
                   /  ___\
           ___     |\/ 5 |
      16*\/ 5 *asin|-----|
                   \  2  /
4*I + --------------------
               5
$$\frac{16 \sqrt{5} \operatorname{asin}{\left(\frac{\sqrt{5}}{2} \right)}}{5} + 4 i$$ 
=
= 
                   /  ___\
           ___     |\/ 5 |
      16*\/ 5 *asin|-----|
                   \  2  /
4*I + --------------------
               5
$$\frac{16 \sqrt{5} \operatorname{asin}{\left(\frac{\sqrt{5}}{2} \right)}}{5} + 4 i$$ 
4*i + 16*sqrt(5)*asin(sqrt(5)/2)/5
Numerical answer [src] 
(11.2396463436482 + 0.557937300164848j)
(11.2396463436482 + 0.557937300164848j)

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

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