Mister Exam

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two x- three /(x- one)(x+2)
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two x minus three divide by (x minus one)(x plus 2)
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2x-3/(x-1)(x+2)dx
Similar expressions
2x-3/(x+1)(x+2)
2x+3/(x-1)(x+2)
2x-3/(x-1)(x-2)

Solving integrals/ 2x-3/(x-1)(x+2)

Integral of 2x-3/(x-1)(x+2) dx

The solution

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  1                         
  /                         
 |                          
 |  /        3          \   
 |  |2*x - -----*(x + 2)| dx
 |  \      x - 1        /   
 |                          
/                           
0

\int\limits_{0}^{1} \left(2 x - \frac{3}{x - 1} \left(x + 2\right)\right)\, dx

Integral(2*x - 3/(x - 1)*(x + 2), (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:
  
  $\int \left(- \frac{3}{x - 1} \left(x + 2\right)\right)\, dx = - \int \frac{3 \left(x + 2\right)}{x - 1}\, dx$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{3 \left(x + 2\right)}{x - 1}\, dx = 3 \int \frac{x + 2}{x - 1}\, dx$
    1. There are multiple ways to do this integral.
      Method #1
      
      Rewrite the integrand:
      
      $\frac{x + 2}{x - 1} = 1 + \frac{3}{x - 1}$
      
      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:
      
      $\int \frac{3}{x - 1}\, dx = 3 \int \frac{1}{x - 1}\, dx$
      
      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: $3 \log{\left(x - 1 \right)}$
      
      The result is: $x + 3 \log{\left(x - 1 \right)}$
      Method #2
      
      Rewrite the integrand:
      
      $\frac{x + 2}{x - 1} = \frac{x}{x - 1} + \frac{2}{x - 1}$
      
      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:
      
      $\int \frac{2}{x - 1}\, dx = 2 \int \frac{1}{x - 1}\, dx$
      
      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: $x + \log{\left(x - 1 \right)} + 2 \log{\left(x - 1 \right)}$
    So, the result is: $3 x + 9 \log{\left(x - 1 \right)}$
  So, the result is: $- 3 x - 9 \log{\left(x - 1 \right)}$
The result is: $x^{2} - 3 x - 9 \log{\left(x - 1 \right)}$
Add the constant of integration:

$x^{2} - 3 x - 9 \log{\left(x - 1 \right)}+ \mathrm{constant}$

The answer is:

x^{2} - 3 x - 9 \log{\left(x - 1 \right)}+ \mathrm{constant}

The answer (Indefinite) [src]

  /                                                       
 |                                                        
 | /        3          \           2                      
 | |2*x - -----*(x + 2)| dx = C + x  - 9*log(-1 + x) - 3*x
 | \      x - 1        /                                  
 |                                                        
/

\int \left(2 x - \frac{3}{x - 1} \left(x + 2\right)\right)\, dx = C + x^{2} - 3 x - 9 \log{\left(x - 1 \right)}

Simplify

The graph

Plot the graph f(x)

The answer [src]

oo + 9*pi*I

\infty + 9 i \pi

oo + 9*pi*I

\infty + 9 i \pi

oo + 9*pi*i

Numerical answer [src]

394.818611075975

394.818611075975

The graph

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

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Identical expressions

Similar expressions

Integral of 2x-3/(x-1)(x+2) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2