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Solving integrals/ 1/sqrt(2-3x^2)

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

The solution

You have entered [src]

  1                   
  /                   
 |                    
 |          1         
 |  1*------------- dx
 |       __________   
 |      /        2    
 |    \/  2 - 3*x     
 |                    
/                     
0

$$\int\limits_{0}^{1} 1 \cdot \frac{1}{\sqrt{2 - 3 x^{2}}}\, dx$$

Integral(1/sqrt(2 - 3*x^2), (x, 0, 1))

Detail solution

TrigSubstitutionRule(theta=_theta, func=sqrt(6)*sin(_theta)/3, rewritten=sqrt(3)/3, substep=ConstantRule(constant=sqrt(3)/3, context=sqrt(3)/3, symbol=_theta), restriction=(x > -sqrt(6)/3) & (x < sqrt(6)/3), context=1/sqrt(2 - 3*x**2), symbol=x)

Now simplify:

$\begin{cases} \frac{\sqrt{3} \operatorname{asin}{\left(\frac{\sqrt{6} x}{2} \right)}}{3} & \text{for}\: x > - \frac{\sqrt{6}}{3} \wedge x < \frac{\sqrt{6}}{3} \\\text{NaN} & \text{otherwise} \end{cases}$
Add the constant of integration:

$\begin{cases} \frac{\sqrt{3} \operatorname{asin}{\left(\frac{\sqrt{6} x}{2} \right)}}{3} & \text{for}\: x > - \frac{\sqrt{6}}{3} \wedge x < \frac{\sqrt{6}}{3} \\\text{NaN} & \text{otherwise} \end{cases}+ \mathrm{constant}$

The answer is:

$\begin{cases} \frac{\sqrt{3} \operatorname{asin}{\left(\frac{\sqrt{6} x}{2} \right)}}{3} & \text{for}\: x > - \frac{\sqrt{6}}{3} \wedge x < \frac{\sqrt{6}}{3} \\\text{NaN} & \text{otherwise} \end{cases}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                         //          /    ___\                                 \
 |                          ||  ___     |x*\/ 6 |                                 |
 |         1                ||\/ 3 *asin|-------|         /       ___         ___\|
 | 1*------------- dx = C + |<          \   2   /         |    -\/ 6        \/ 6 ||
 |      __________          ||-------------------  for And|x > -------, x < -----||
 |     /        2           ||         3                  \       3           3  /|
 |   \/  2 - 3*x            \\                                                    /
 |                                                                                 
/

$$\int 1 \cdot \frac{1}{\sqrt{2 - 3 x^{2}}}\, dx = C + \begin{cases} \frac{\sqrt{3} \operatorname{asin}{\left(\frac{\sqrt{6} x}{2} \right)}}{3} & \text{for}\: x > - \frac{\sqrt{6}}{3} \wedge x < \frac{\sqrt{6}}{3} \end{cases}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

          /  ___\
  ___     |\/ 6 |
\/ 3 *asin|-----|
          \  2  /
-----------------
        3

$$\frac{\sqrt{3} \operatorname{asin}{\left(\frac{\sqrt{6}}{2} \right)}}{3}$$

          /  ___\
  ___     |\/ 6 |
\/ 3 *asin|-----|
          \  2  /
-----------------
        3

$$\frac{\sqrt{3} \operatorname{asin}{\left(\frac{\sqrt{6}}{2} \right)}}{3}$$

Упростить

Numerical answer [src]

(0.897124945273862 - 0.344125471092143j)

(0.897124945273862 - 0.344125471092143j)

The graph

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

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Identical expressions

Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution