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Identical expressions
(x^3dx)/(sqrt(x- one))
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x^3dx/sqrtx-1
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Similar expressions
(x^3dx)/(sqrt(x+1))

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

The solution

You have entered [src]

  1             
  /             
 |              
 |       3      
 |      x       
 |  --------- dx
 |    _______   
 |  \/ x - 1    
 |              
/               
0

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

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

Detail solution

Let $u = \sqrt{x - 1}$ .

Then let $du = \frac{dx}{2 \sqrt{x - 1}}$ and substitute $2 du$ :

$\int 2 \left(u^{2} + 1\right)^{3}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(u^{2} + 1\right)^{3}\, du = 2 \int \left(u^{2} + 1\right)^{3}\, du$
  1. Rewrite the integrand:
    
    $\left(u^{2} + 1\right)^{3} = u^{6} + 3 u^{4} + 3 u^{2} + 1$
  2. Integrate term-by-term:
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{6}\, du = \frac{u^{7}}{7}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 3 u^{4}\, du = 3 \int u^{4}\, du$
      1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
        
        $\int u^{4}\, du = \frac{u^{5}}{5}$
      So, the result is: $\frac{3 u^{5}}{5}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 3 u^{2}\, du = 3 \int u^{2}\, du$
      1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
        
        $\int u^{2}\, du = \frac{u^{3}}{3}$
      So, the result is: $u^{3}$
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
    The result is: $\frac{u^{7}}{7} + \frac{3 u^{5}}{5} + u^{3} + u$
  So, the result is: $\frac{2 u^{7}}{7} + \frac{6 u^{5}}{5} + 2 u^{3} + 2 u$
Now substitute $u$ back in:

$\frac{2 \left(x - 1\right)^{\frac{7}{2}}}{7} + \frac{6 \left(x - 1\right)^{\frac{5}{2}}}{5} + 2 \left(x - 1\right)^{\frac{3}{2}} + 2 \sqrt{x - 1}$
Now simplify:

$\frac{2 \sqrt{x - 1} \left(5 x^{3} + 6 x^{2} + 8 x + 16\right)}{35}$
Add the constant of integration:

$\frac{2 \sqrt{x - 1} \left(5 x^{3} + 6 x^{2} + 8 x + 16\right)}{35}+ \mathrm{constant}$

The answer is:

$\frac{2 \sqrt{x - 1} \left(5 x^{3} + 6 x^{2} + 8 x + 16\right)}{35}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                           
 |                                                                            
 |      3                                                   7/2            5/2
 |     x                  _______            3/2   2*(x - 1)      6*(x - 1)   
 | --------- dx = C + 2*\/ x - 1  + 2*(x - 1)    + ------------ + ------------
 |   _______                                            7              5      
 | \/ x - 1                                                                   
 |                                                                            
/

$$\int \frac{x^{3}}{\sqrt{x - 1}}\, dx = C + \frac{2 \left(x - 1\right)^{\frac{7}{2}}}{7} + \frac{6 \left(x - 1\right)^{\frac{5}{2}}}{5} + 2 \left(x - 1\right)^{\frac{3}{2}} + 2 \sqrt{x - 1}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

-32*I
-----
  35

$$- \frac{32 i}{35}$$

-32*I
-----
  35

$$- \frac{32 i}{35}$$

-32*i/35

Numerical answer [src]

(0.0 - 0.914285713615916j)

(0.0 - 0.914285713615916j)

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

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

Limits of integration:

The graph:

Piecewise:

The solution