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Identical expressions
(x- one)/((x- two)^(five / three))
(x minus 1) divide by ((x minus 2) to the power of (5 divide by 3))
(x minus one) divide by ((x minus two) to the power of (five divide by three))
(x-1)/((x-2)(5/3))
x-1/x-25/3
x-1/x-2^5/3
(x-1) divide by ((x-2)^(5 divide by 3))
(x-1)/((x-2)^(5/3))dx
Similar expressions
(x+1)/((x-2)^(5/3))
(x-1)/((x+2)^(5/3))

Integral of (x-1)/((x-2)^(5/3)) dx

The solution

You have entered [src]

  3              
  /              
 |               
 |    x - 1      
 |  ---------- dx
 |         5/3   
 |  (x - 2)      
 |               
/                
1

$$\int\limits_{1}^{3} \frac{x - 1}{\left(x - 2\right)^{\frac{5}{3}}}\, dx$$

Integral((x - 1)/(x - 2)^(5/3), (x, 1, 3))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\frac{x - 1}{\left(x - 2\right)^{\frac{5}{3}}} = \frac{x - 1}{x \left(x - 2\right)^{\frac{2}{3}} - 2 \left(x - 2\right)^{\frac{2}{3}}}$
2. Rewrite the integrand:
  
  $\frac{x - 1}{x \left(x - 2\right)^{\frac{2}{3}} - 2 \left(x - 2\right)^{\frac{2}{3}}} = \frac{x}{x \left(x - 2\right)^{\frac{2}{3}} - 2 \left(x - 2\right)^{\frac{2}{3}}} - \frac{1}{x \left(x - 2\right)^{\frac{2}{3}} - 2 \left(x - 2\right)^{\frac{2}{3}}}$
3. Integrate term-by-term:
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $\int \frac{x}{\left(x - 2\right)^{\frac{5}{3}}}\, dx$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{1}{x \left(x - 2\right)^{\frac{2}{3}} - 2 \left(x - 2\right)^{\frac{2}{3}}}\right)\, dx = - \int \frac{1}{x \left(x - 2\right)^{\frac{2}{3}} - 2 \left(x - 2\right)^{\frac{2}{3}}}\, dx$
    1. Don't know the steps in finding this integral.
      
      But the integral is
      
      $\int \frac{1}{x \left(x - 2\right)^{\frac{2}{3}} - 2 \left(x - 2\right)^{\frac{2}{3}}}\, dx$
    So, the result is: $- \int \frac{1}{x \left(x - 2\right)^{\frac{2}{3}} - 2 \left(x - 2\right)^{\frac{2}{3}}}\, dx$
  The result is: $\int \frac{x}{\left(x - 2\right)^{\frac{5}{3}}}\, dx - \int \frac{1}{x \left(x - 2\right)^{\frac{2}{3}} - 2 \left(x - 2\right)^{\frac{2}{3}}}\, dx$
Method #2
1. Rewrite the integrand:
  
  $\frac{x - 1}{\left(x - 2\right)^{\frac{5}{3}}} = \frac{x}{x \left(x - 2\right)^{\frac{2}{3}} - 2 \left(x - 2\right)^{\frac{2}{3}}} - \frac{1}{x \left(x - 2\right)^{\frac{2}{3}} - 2 \left(x - 2\right)^{\frac{2}{3}}}$
2. Integrate term-by-term:
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $\int \frac{x}{\left(x - 2\right)^{\frac{5}{3}}}\, dx$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{1}{x \left(x - 2\right)^{\frac{2}{3}} - 2 \left(x - 2\right)^{\frac{2}{3}}}\right)\, dx = - \int \frac{1}{x \left(x - 2\right)^{\frac{2}{3}} - 2 \left(x - 2\right)^{\frac{2}{3}}}\, dx$
    1. Don't know the steps in finding this integral.
      
      But the integral is
      
      $\int \frac{1}{x \left(x - 2\right)^{\frac{2}{3}} - 2 \left(x - 2\right)^{\frac{2}{3}}}\, dx$
    So, the result is: $- \int \frac{1}{x \left(x - 2\right)^{\frac{2}{3}} - 2 \left(x - 2\right)^{\frac{2}{3}}}\, dx$
  The result is: $\int \frac{x}{\left(x - 2\right)^{\frac{5}{3}}}\, dx - \int \frac{1}{x \left(x - 2\right)^{\frac{2}{3}} - 2 \left(x - 2\right)^{\frac{2}{3}}}\, dx$
Now simplify:

$\frac{3 x}{\left(x - 2\right)^{\frac{2}{3}}} - \frac{15}{2 \left(x - 2\right)^{\frac{2}{3}}}$
Add the constant of integration:

$\frac{3 x}{\left(x - 2\right)^{\frac{2}{3}}} - \frac{15}{2 \left(x - 2\right)^{\frac{2}{3}}}+ \mathrm{constant}$

The answer is:

$\frac{3 x}{\left(x - 2\right)^{\frac{2}{3}}} - \frac{15}{2 \left(x - 2\right)^{\frac{2}{3}}}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                      /                                       /              
 |                      |                                       |               
 |   x - 1              |                1                      |      x        
 | ---------- dx = C -  | ------------------------------- dx +  | ----------- dx
 |        5/3           |             2/3             2/3       |         5/3   
 | (x - 2)              | - 2*(-2 + x)    + x*(-2 + x)          | (-2 + x)      
 |                      |                                       |               
/                      /                                       /

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

                         3 ____
            /3 ____\   9*\/ -1 
oo + oo*sign\\/ -1 / - --------
                          2

$$\infty - \frac{9 \sqrt[3]{-1}}{2} + \infty \operatorname{sign}{\left(\sqrt[3]{-1} \right)}$$

                         3 ____
            /3 ____\   9*\/ -1 
oo + oo*sign\\/ -1 / - --------
                          2

$$\infty - \frac{9 \sqrt[3]{-1}}{2} + \infty \operatorname{sign}{\left(\sqrt[3]{-1} \right)}$$

oo + oo*sign((-1)^(1/3)) - 9*(-1)^(1/3)/2

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

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Identical expressions

Similar expressions

Integral of (x-1)/((x-2)^(5/3)) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2