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Similar expressions
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Integral of 1-root3(2x)/sqrt(2x) dx

The solution

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

$$\int\limits_{0}^{1} \left(1 - \frac{\sqrt[3]{2 x}}{\sqrt{2 x}}\right)\, dx$$

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

Detail solution

Integrate term-by-term:
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 1\, dx = x$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{\sqrt[3]{2 x}}{\sqrt{2 x}}\right)\, dx = - \int \frac{\sqrt[3]{2 x}}{\sqrt{2 x}}\, dx$
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $\begin{cases} \frac{2^{\frac{5}{6}} x^{\frac{5}{6}} \Gamma\left(\frac{5}{6}\right)}{2 \Gamma\left(\frac{11}{6}\right)} & \text{for}\: \left|{x}\right| > 1 \vee \left|{x}\right| < 1 \\\frac{2^{\frac{5}{6}} {G_{2, 2}^{1, 1}\left(\begin{matrix} 1 & \frac{11}{6} \\\frac{5}{6} & 0 \end{matrix} \middle| {x} \right)}}{2} + \frac{2^{\frac{5}{6}} {G_{2, 2}^{0, 2}\left(\begin{matrix} \frac{11}{6}, 1 & \\ & \frac{5}{6}, 0 \end{matrix} \middle| {x} \right)}}{2} & \text{otherwise} \end{cases}$
  So, the result is: $- \begin{cases} \frac{2^{\frac{5}{6}} x^{\frac{5}{6}} \Gamma\left(\frac{5}{6}\right)}{2 \Gamma\left(\frac{11}{6}\right)} & \text{for}\: \left|{x}\right| > 1 \vee \left|{x}\right| < 1 \\\frac{2^{\frac{5}{6}} {G_{2, 2}^{1, 1}\left(\begin{matrix} 1 & \frac{11}{6} \\\frac{5}{6} & 0 \end{matrix} \middle| {x} \right)}}{2} + \frac{2^{\frac{5}{6}} {G_{2, 2}^{0, 2}\left(\begin{matrix} \frac{11}{6}, 1 & \\ & \frac{5}{6}, 0 \end{matrix} \middle| {x} \right)}}{2} & \text{otherwise} \end{cases}$
The result is: $x - \begin{cases} \frac{2^{\frac{5}{6}} x^{\frac{5}{6}} \Gamma\left(\frac{5}{6}\right)}{2 \Gamma\left(\frac{11}{6}\right)} & \text{for}\: \left|{x}\right| > 1 \vee \left|{x}\right| < 1 \\\frac{2^{\frac{5}{6}} {G_{2, 2}^{1, 1}\left(\begin{matrix} 1 & \frac{11}{6} \\\frac{5}{6} & 0 \end{matrix} \middle| {x} \right)}}{2} + \frac{2^{\frac{5}{6}} {G_{2, 2}^{0, 2}\left(\begin{matrix} \frac{11}{6}, 1 & \\ & \frac{5}{6}, 0 \end{matrix} \middle| {x} \right)}}{2} & \text{otherwise} \end{cases}$
Now simplify:

$\begin{cases} - \frac{3 \cdot 2^{\frac{5}{6}} x^{\frac{5}{6}}}{5} + x & \text{for}\: \left|{x}\right| \neq 1 \\x - \frac{2^{\frac{5}{6}} \left({G_{2, 2}^{1, 1}\left(\begin{matrix} 1 & \frac{11}{6} \\\frac{5}{6} & 0 \end{matrix} \middle| {x} \right)} + {G_{2, 2}^{0, 2}\left(\begin{matrix} \frac{11}{6}, 1 & \\ & \frac{5}{6}, 0 \end{matrix} \middle| {x} \right)}\right)}{2} & \text{otherwise} \end{cases}$
Add the constant of integration:

$\begin{cases} - \frac{3 \cdot 2^{\frac{5}{6}} x^{\frac{5}{6}}}{5} + x & \text{for}\: \left|{x}\right| \neq 1 \\x - \frac{2^{\frac{5}{6}} \left({G_{2, 2}^{1, 1}\left(\begin{matrix} 1 & \frac{11}{6} \\\frac{5}{6} & 0 \end{matrix} \middle| {x} \right)} + {G_{2, 2}^{0, 2}\left(\begin{matrix} \frac{11}{6}, 1 & \\ & \frac{5}{6}, 0 \end{matrix} \middle| {x} \right)}\right)}{2} & \text{otherwise} \end{cases}+ \mathrm{constant}$

The answer is:

$\begin{cases} - \frac{3 \cdot 2^{\frac{5}{6}} x^{\frac{5}{6}}}{5} + x & \text{for}\: \left|{x}\right| \neq 1 \\x - \frac{2^{\frac{5}{6}} \left({G_{2, 2}^{1, 1}\left(\begin{matrix} 1 & \frac{11}{6} \\\frac{5}{6} & 0 \end{matrix} \middle| {x} \right)} + {G_{2, 2}^{0, 2}\left(\begin{matrix} \frac{11}{6}, 1 & \\ & \frac{5}{6}, 0 \end{matrix} \middle| {x} \right)}\right)}{2} & \text{otherwise} \end{cases}+ \mathrm{constant}$

The answer (Indefinite) [src]

                              //                       5/6  5/6                                                            \
  /                           ||                      2   *x   *Gamma(5/6)                                                 |
 |                            ||                      --------------------                         for Or(|x| > 1, |x| < 1)|
 | /    3 _____\              ||                         2*Gamma(11/6)                                                     |
 | |    \/ 2*x |              ||                                                                                           |
 | |1 - -------| dx = C + x - |< 5/6  __1, 1 / 1   11/6 |  \    5/6  __0, 2 /11/6, 1         |  \                          |
 | |      _____|              ||2   */__     |          | x|   2   */__     |                | x|                          |
 | \    \/ 2*x /              ||     \_|2, 2 \5/6   0   |  /        \_|2, 2 \         5/6, 0 |  /                          |
 |                            ||---------------------------- + ----------------------------------         otherwise        |
/                             ||             2                                 2                                           |
                              \\                                                                                           /

$$\int \left(1 - \frac{\sqrt[3]{2 x}}{\sqrt{2 x}}\right)\, dx = C + x - \begin{cases} \frac{2^{\frac{5}{6}} x^{\frac{5}{6}} \Gamma\left(\frac{5}{6}\right)}{2 \Gamma\left(\frac{11}{6}\right)} & \text{for}\: \left|{x}\right| > 1 \vee \left|{x}\right| < 1 \\\frac{2^{\frac{5}{6}} {G_{2, 2}^{1, 1}\left(\begin{matrix} 1 & \frac{11}{6} \\\frac{5}{6} & 0 \end{matrix} \middle| {x} \right)}}{2} + \frac{2^{\frac{5}{6}} {G_{2, 2}^{0, 2}\left(\begin{matrix} \frac{11}{6}, 1 & \\ & \frac{5}{6}, 0 \end{matrix} \middle| {x} \right)}}{2} & \text{otherwise} \end{cases}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

       5/6
    3*2   
1 - ------
      5

$$1 - \frac{3 \cdot 2^{\frac{5}{6}}}{5}$$

       5/6
    3*2   
1 - ------
      5

$$1 - \frac{3 \cdot 2^{\frac{5}{6}}}{5}$$

1 - 3*2^(5/6)/5

Numerical answer [src]

-0.069078461768407

-0.069078461768407

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

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Identical expressions

Similar expressions

Integral of 1-root3(2x)/sqrt(2x) dx

Limits of integration:

The graph:

Piecewise:

The solution