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Identical expressions
x^ two /(x^ two - five *x+ six)
x squared divide by (x squared minus 5 multiply by x plus 6)
x to the power of two divide by (x to the power of two minus five multiply by x plus six)
x2/(x2-5*x+6)
x2/x2-5*x+6
x²/(x²-5*x+6)
x to the power of 2/(x to the power of 2-5*x+6)
x^2/(x^2-5x+6)
x2/(x2-5x+6)
x2/x2-5x+6
x^2/x^2-5x+6
x^2 divide by (x^2-5*x+6)
x^2/(x^2-5*x+6)dx
Similar expressions
x^2/(x^2+5*x+6)
x^2/(x^2-5*x-6)

Solving integrals/ x^2/(x^2-5*x+6)

Integral of x^2/(x^2-5*x+6) dx

The solution

You have entered [src]

  1                
  /                
 |                 
 |        2        
 |       x         
 |  ------------ dx
 |   2             
 |  x  - 5*x + 6   
 |                 
/                  
0

$$\int\limits_{0}^{1} \frac{x^{2}}{\left(x^{2} - 5 x\right) + 6}\, dx$$

Integral(x^2/(x^2 - 5*x + 6), (x, 0, 1))

Detail solution

Rewrite the integrand:

$\frac{x^{2}}{\left(x^{2} - 5 x\right) + 6} = 1 - \frac{4}{x - 2} + \frac{9}{x - 3}$
Integrate term-by-term:
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 1\, dx = x$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{4}{x - 2}\right)\, dx = - 4 \int \frac{1}{x - 2}\, dx$
  1. Let $u = x - 2$ .
    
    Then let $du = dx$ 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:
    
    $\log{\left(x - 2 \right)}$
  So, the result is: $- 4 \log{\left(x - 2 \right)}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{9}{x - 3}\, dx = 9 \int \frac{1}{x - 3}\, dx$
  1. Let $u = x - 3$ .
    
    Then let $du = dx$ 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:
    
    $\log{\left(x - 3 \right)}$
  So, the result is: $9 \log{\left(x - 3 \right)}$
The result is: $x + 9 \log{\left(x - 3 \right)} - 4 \log{\left(x - 2 \right)}$
Add the constant of integration:

$x + 9 \log{\left(x - 3 \right)} - 4 \log{\left(x - 2 \right)}+ \mathrm{constant}$

The answer is:

$x + 9 \log{\left(x - 3 \right)} - 4 \log{\left(x - 2 \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                       
 |                                                        
 |       2                                                
 |      x                                                 
 | ------------ dx = C + x - 4*log(-2 + x) + 9*log(-3 + x)
 |  2                                                     
 | x  - 5*x + 6                                           
 |                                                        
/

$$\int \frac{x^{2}}{\left(x^{2} - 5 x\right) + 6}\, dx = C + x + 9 \log{\left(x - 3 \right)} - 4 \log{\left(x - 2 \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

1 - 9*log(3) + 13*log(2)

$$- 9 \log{\left(3 \right)} + 1 + 13 \log{\left(2 \right)}$$

1 - 9*log(3) + 13*log(2)

$$- 9 \log{\left(3 \right)} + 1 + 13 \log{\left(2 \right)}$$

1 - 9*log(3) + 13*log(2)

Numerical answer [src]

0.123402749266302

0.123402749266302

The graph

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

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Identical expressions

Similar expressions

Integral of x^2/(x^2-5*x+6) dx

Limits of integration:

The graph:

Piecewise:

The solution