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Solving integrals/ /sqrt(5-3x^2)

Integral of /sqrt(5-3x^2) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1                 
  /                 
 |                  
 |     __________   
 |    /        2    
 |  \/  5 - 3*x   dx
 |                  
/                   
0
$$\int\limits_{0}^{1} \sqrt{5 - 3 x^{2}}\, dx$$ 
Integral(sqrt(5 - 3*x^2), (x, 0, 1))
Detail solution

    TrigSubstitutionRule(theta=_theta, func=sqrt(15)*sin(_theta)/3, rewritten=5*sqrt(3)*cos(_theta)**2/3, substep=ConstantTimesRule(constant=5*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=5*sqrt(3)*cos(_theta)**2/3, symbol=_theta), restriction=(x > -sqrt(15)/3) & (x < sqrt(15)/3), context=sqrt(5 - 3*x**2), symbol=x)

  1. Now simplify:

  2. Add the constant of integration:

The answer is:

The answer (Indefinite) [src] 
  /                       //        /    /    ____\                        \                                   \
 |                        ||        |    |x*\/ 15 |              __________|                                   |
 |    __________          ||        |asin|--------|       ___   /        2 |                                   |
 |   /        2           ||    ___ |    \   5    /   x*\/ 3 *\/  5 - 3*x  |                                   |
 | \/  5 - 3*x   dx = C + |<5*\/ 3 *|-------------- + ---------------------|         /       ____         ____\|
 |                        ||        \      2                    10         /         |    -\/ 15        \/ 15 ||
/                         ||------------------------------------------------  for And|x > --------, x < ------||
                          ||                       3                                 \       3            3   /|
                          \\                                                                                   /
$$\int \sqrt{5 - 3 x^{2}}\, dx = C + \begin{cases} \frac{5 \sqrt{3} \left(\frac{\sqrt{3} x \sqrt{5 - 3 x^{2}}}{10} + \frac{\operatorname{asin}{\left(\frac{\sqrt{15} x}{5} \right)}}{2}\right)}{3} & \text{for}\: x > - \frac{\sqrt{15}}{3} \wedge x < \frac{\sqrt{15}}{3} \end{cases}$$
Simplify
The graph
Plot the graph f(x)
The answer [src] 
                    /  ____\
            ___     |\/ 15 |
  ___   5*\/ 3 *asin|------|
\/ 2                \  5   /
----- + --------------------
  2              6
$$\frac{\sqrt{2}}{2} + \frac{5 \sqrt{3} \operatorname{asin}{\left(\frac{\sqrt{15}}{5} \right)}}{6}$$ 
=
= 
                    /  ____\
            ___     |\/ 15 |
  ___   5*\/ 3 *asin|------|
\/ 2                \  5   /
----- + --------------------
  2              6
$$\frac{\sqrt{2}}{2} + \frac{5 \sqrt{3} \operatorname{asin}{\left(\frac{\sqrt{15}}{5} \right)}}{6}$$ 
sqrt(2)/2 + 5*sqrt(3)*asin(sqrt(15)/5)/6
Numerical answer [src] 
1.98604894604763
1.98604894604763

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

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