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Identical expressions
(sixteen - three (x)^ two)^ one / two
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(16-3(x)^2)^1/2dx
Similar expressions
(16+3(x)^2)^1/2

Integral of (16-3(x)^2)^1/2 dx

The solution

You have entered [src]

  2                  
  /                  
 |                   
 |     ___________   
 |    /         2    
 |  \/  16 - 3*x   dx
 |                   
/                    
-2

$$\int\limits_{-2}^{2} \sqrt{16 - 3 x^{2}}\, dx$$

Integral(sqrt(16 - 3*x^2), (x, -2, 2))

Detail solution

TrigSubstitutionRule(theta=_theta, func=4*sqrt(3)*sin(_theta)/3, rewritten=16*sqrt(3)*cos(_theta)**2/3, substep=ConstantTimesRule(constant=16*sqrt(3)/3, 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(3)*cos(_theta)**2/3, symbol=_theta), restriction=(x > -4*sqrt(3)/3) & (x < 4*sqrt(3)/3), context=sqrt(16 - 3*x**2), symbol=x)

Now simplify:

$\begin{cases} \frac{x \sqrt{16 - 3 x^{2}}}{2} + \frac{8 \sqrt{3} \operatorname{asin}{\left(\frac{\sqrt{3} x}{4} \right)}}{3} & \text{for}\: x > - \frac{4 \sqrt{3}}{3} \wedge x < \frac{4 \sqrt{3}}{3} \end{cases}$
Add the constant of integration:

$\begin{cases} \frac{x \sqrt{16 - 3 x^{2}}}{2} + \frac{8 \sqrt{3} \operatorname{asin}{\left(\frac{\sqrt{3} x}{4} \right)}}{3} & \text{for}\: x > - \frac{4 \sqrt{3}}{3} \wedge x < \frac{4 \sqrt{3}}{3} \end{cases}+ \mathrm{constant}$

The answer is:

$\begin{cases} \frac{x \sqrt{16 - 3 x^{2}}}{2} + \frac{8 \sqrt{3} \operatorname{asin}{\left(\frac{\sqrt{3} x}{4} \right)}}{3} & \text{for}\: x > - \frac{4 \sqrt{3}}{3} \wedge x < \frac{4 \sqrt{3}}{3} \end{cases}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                        //         /    /    ___\                         \                                    \
 |                         ||         |    |x*\/ 3 |              ___________|                                    |
 |    ___________          ||         |asin|-------|       ___   /         2 |                                    |
 |   /         2           ||     ___ |    \   4   /   x*\/ 3 *\/  16 - 3*x  |                                    |
 | \/  16 - 3*x   dx = C + |<16*\/ 3 *|------------- + ----------------------|         /         ___          ___\|
 |                         ||         \      2                   32          /         |    -4*\/ 3       4*\/ 3 ||
/                          ||-------------------------------------------------  for And|x > --------, x < -------||
                           ||                        3                                 \       3             3   /|
                           \\                                                                                     /

$$\int \sqrt{16 - 3 x^{2}}\, dx = C + \begin{cases} \frac{16 \sqrt{3} \left(\frac{\sqrt{3} x \sqrt{16 - 3 x^{2}}}{32} + \frac{\operatorname{asin}{\left(\frac{\sqrt{3} x}{4} \right)}}{2}\right)}{3} & \text{for}\: x > - \frac{4 \sqrt{3}}{3} \wedge x < \frac{4 \sqrt{3}}{3} \end{cases}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

            ___
    16*pi*\/ 3 
4 + -----------
         9

$$4 + \frac{16 \sqrt{3} \pi}{9}$$

            ___
    16*pi*\/ 3 
4 + -----------
         9

$$4 + \frac{16 \sqrt{3} \pi}{9}$$

4 + 16*pi*sqrt(3)/9

Numerical answer [src]

13.6735966092492

13.6735966092492

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

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Integral of (16-3(x)^2)^1/2 dx

Limits of integration:

The graph:

Piecewise:

The solution