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

The solution

You have entered [src]

  1             
  /             
 |              
 |       2      
 |      x       
 |  --------- dx
 |    ___       
 |  \/ x  - 1   
 |              
/               
0

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

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

Detail solution

Let $u = \sqrt{x}$ .

Then let $du = \frac{dx}{2 \sqrt{x}}$ and substitute $2 du$ :

$\int \frac{2 u^{5}}{u - 1}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{u^{5}}{u - 1}\, du = 2 \int \frac{u^{5}}{u - 1}\, du$
  1. Rewrite the integrand:
    
    $\frac{u^{5}}{u - 1} = u^{4} + u^{3} + u^{2} + u + 1 + \frac{1}{u - 1}$
  2. Integrate term-by-term:
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{4}\, du = \frac{u^{5}}{5}$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{3}\, du = \frac{u^{4}}{4}$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
    1. Let $u = u - 1$ .
      
      Then let $du = du$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      Now substitute $u$ back in:
      
      $\log{\left(u - 1 \right)}$
    The result is: $\frac{u^{5}}{5} + \frac{u^{4}}{4} + \frac{u^{3}}{3} + \frac{u^{2}}{2} + u + \log{\left(u - 1 \right)}$
  So, the result is: $\frac{2 u^{5}}{5} + \frac{u^{4}}{2} + \frac{2 u^{3}}{3} + u^{2} + 2 u + 2 \log{\left(u - 1 \right)}$
Now substitute $u$ back in:

$\frac{2 x^{\frac{5}{2}}}{5} + \frac{2 x^{\frac{3}{2}}}{3} + 2 \sqrt{x} + \frac{x^{2}}{2} + x + 2 \log{\left(\sqrt{x} - 1 \right)}$
Add the constant of integration:

$\frac{2 x^{\frac{5}{2}}}{5} + \frac{2 x^{\frac{3}{2}}}{3} + 2 \sqrt{x} + \frac{x^{2}}{2} + x + 2 \log{\left(\sqrt{x} - 1 \right)}+ \mathrm{constant}$

The answer is:

$\frac{2 x^{\frac{5}{2}}}{5} + \frac{2 x^{\frac{3}{2}}}{3} + 2 \sqrt{x} + \frac{x^{2}}{2} + x + 2 \log{\left(\sqrt{x} - 1 \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                         
 |                                                                          
 |      2                  2                                    3/2      5/2
 |     x                  x        ___        /       ___\   2*x      2*x   
 | --------- dx = C + x + -- + 2*\/ x  + 2*log\-1 + \/ x / + ------ + ------
 |   ___                  2                                    3        5   
 | \/ x  - 1                                                                
 |                                                                          
/

$$\int \frac{x^{2}}{\sqrt{x} - 1}\, dx = C + \frac{2 x^{\frac{5}{2}}}{5} + \frac{2 x^{\frac{3}{2}}}{3} + 2 \sqrt{x} + \frac{x^{2}}{2} + x + 2 \log{\left(\sqrt{x} - 1 \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

-oo - 2*pi*I

$$-\infty - 2 i \pi$$

-oo - 2*pi*I

$$-\infty - 2 i \pi$$

-oo - 2*pi*i

Numerical answer [src]

-85.0014007435013

-85.0014007435013

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

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Identical expressions

Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution