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Identical expressions
x^(three / two)/sqrt(two)
x to the power of (3 divide by 2) divide by square root of (2)
x to the power of (three divide by two) divide by square root of (two)
x^(3/2)/√(2)
x(3/2)/sqrt(2)
x3/2/sqrt2
x^3/2/sqrt2
x^(3 divide by 2) divide by sqrt(2)
x^(3/2)/sqrt(2)dx

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

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

The solution

You have entered [src]

  1         
  /         
 |          
 |    3/2   
 |   x      
 |  ----- dx
 |    ___   
 |  \/ 2    
 |          
/           
1/2

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

Integral(x^(3/2)/(sqrt(2)), (x, 1/2, 1))

Detail solution

The integral of a constant times a function is the constant times the integral of the function:

$\int \frac{x^{\frac{3}{2}}}{\sqrt{2}}\, dx = \frac{\sqrt{2}}{2} \int x^{\frac{3}{2}}\, dx$
1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int x^{\frac{3}{2}}\, dx = \frac{2 x^{\frac{5}{2}}}{5}$
So, the result is: $\frac{2 \frac{\sqrt{2}}{2} x^{\frac{5}{2}}}{5}$
Now simplify:

$\frac{\sqrt{2} x^{\frac{5}{2}}}{5}$
Add the constant of integration:

$\frac{\sqrt{2} x^{\frac{5}{2}}}{5}+ \mathrm{constant}$

The answer is:

$\frac{\sqrt{2} x^{\frac{5}{2}}}{5}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                        ___
 |                   5/2 \/ 2 
 |   3/2          2*x   *-----
 |  x                      2  
 | ----- dx = C + ------------
 |   ___               5      
 | \/ 2                       
 |                            
/

$${{\sqrt{2}\,x^{{{5}\over{2}}}}\over{5}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

         ___
  1    \/ 2 
- -- + -----
  20     5

$${{{{2}\over{5}}-{{1}\over{5\,2^{{{3}\over{2}}}}}}\over{\sqrt{2}}}$$

         ___
  1    \/ 2 
- -- + -----
  20     5

$$- \frac{1}{20} + \frac{\sqrt{2}}{5}$$

Упростить

Numerical answer [src]

0.232842712474619

0.232842712474619

The graph

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

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Identical expressions

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

Limits of integration:

The graph:

Piecewise:

The solution