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

The solution

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  1                           
  /                           
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 |  /          1          \   
 |  |2*x - 1*-----*(x - 2)| dx
 |  \        x - 1        /   
 |                            
/                             
0

$$\int\limits_{0}^{1} \left(2 x - 1 \cdot \frac{1}{x - 1} \left(x - 2\right)\right)\, dx$$

Integral(2*x - (x - 1*2)/(x - 1*1), (x, 0, 1))

Detail solution

Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 2 x\, dx = 2 \int x\, dx$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x\, dx = \frac{x^{2}}{2}$
  So, the result is: $x^{2}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{1 \left(x - 2\right)}{x - 1}\right)\, dx = - \int \frac{x - 2}{x - 1}\, dx$
  1. There are multiple ways to do this integral.
    Method #1
    1. Rewrite the integrand:
      
      $\frac{x - 2}{x - 1} = 1 - \frac{1}{x - 1}$
    2. Integrate term-by-term:
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, dx = x$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{x - 1}\right)\, dx = - \int \frac{1}{x - 1}\, dx$
      
      Let $u = x - 1$ .
      
      Then let $du = dx$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(x - 1 \right)}$
      
      So, the result is: $- \log{\left(x - 1 \right)}$
      
      The result is: $x - \log{\left(x - 1 \right)}$
    Method #2
    1. Rewrite the integrand:
      
      $\frac{x - 2}{x - 1} = \frac{x}{x - 1} - \frac{2}{x - 1}$
    2. Integrate term-by-term:
      
      Rewrite the integrand:
      
      $\frac{x}{x - 1} = 1 + \frac{1}{x - 1}$
      
      Integrate term-by-term:
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, dx = x$
      
      Let $u = x - 1$ .
      
      Then let $du = dx$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(x - 1 \right)}$
      
      The result is: $x + \log{\left(x - 1 \right)}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{2}{x - 1}\right)\, dx = - 2 \int \frac{1}{x - 1}\, dx$
      
      Let $u = x - 1$ .
      
      Then let $du = dx$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(x - 1 \right)}$
      
      So, the result is: $- 2 \log{\left(x - 1 \right)}$
      
      The result is: $x - \log{\left(x - 1 \right)}$
  So, the result is: $- x + \log{\left(x - 1 \right)}$
The result is: $x^{2} - x + \log{\left(x - 1 \right)}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                                                     
 |                                                      
 | /          1          \           2                  
 | |2*x - 1*-----*(x - 2)| dx = C + x  - x + log(-1 + x)
 | \        x - 1        /                              
 |                                                      
/

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

Simplify

The answer [src]

-oo - pi*I

$$-\infty - i \pi$$

-oo - pi*I

$$-\infty - i \pi$$

Упростить

Numerical answer [src]

-44.0909567862195

-44.0909567862195

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

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

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2