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Identical expressions
two (x^ two +9x+ six)/(x− three)(x^ two +4x+ seven)
2(x squared plus 9x plus 6) divide by (x−3)(x squared plus 4x plus 7)
two (x to the power of two plus 9x plus six) divide by (x− three)(x to the power of two plus 4x plus seven)
2(x2+9x+6)/(x−3)(x2+4x+7)
2x2+9x+6/x−3x2+4x+7
2(x²+9x+6)/(x−3)(x²+4x+7)
2(x to the power of 2+9x+6)/(x−3)(x to the power of 2+4x+7)
2x^2+9x+6/x−3x^2+4x+7
2(x^2+9x+6) divide by (x−3)(x^2+4x+7)
2(x^2+9x+6)/(x−3)(x^2+4x+7)dx
Similar expressions
2(x^2+9x+6)/(x−3)(x^2+4x-7)
2(x^2+9x-6)/(x−3)(x^2+4x+7)
2(x^2-9x+6)/(x−3)(x^2+4x+7)
2(x^2+9x+6)/(x−3)(x^2-4x+7)

Solving integrals/ 2(x^2+9x+6)/(x−3)(x^2+4x+7)

Integral of 2(x^2+9x+6)/(x−3)(x^2+4x+7) dx

The solution

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  1                                   
  /                                   
 |                                    
 |    / 2          \                  
 |  2*\x  + 9*x + 6/ / 2          \   
 |  ----------------*\x  + 4*x + 7/ dx
 |       x - 3                        
 |                                    
/                                     
0

$$\int\limits_{0}^{1} \frac{2 \left(\left(x^{2} + 9 x\right) + 6\right)}{x - 3} \left(\left(x^{2} + 4 x\right) + 7\right)\, dx$$

Integral(((2*(x^2 + 9*x + 6))/(x - 3))*(x^2 + 4*x + 7), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\frac{2 \left(\left(x^{2} + 9 x\right) + 6\right)}{x - 3} \left(\left(x^{2} + 4 x\right) + 7\right) = 2 x^{3} + 32 x^{2} + 194 x + 756 + \frac{2352}{x - 3}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 2 x^{3}\, dx = 2 \int x^{3}\, dx$
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x^{3}\, dx = \frac{x^{4}}{4}$
    So, the result is: $\frac{x^{4}}{2}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 32 x^{2}\, dx = 32 \int x^{2}\, dx$
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x^{2}\, dx = \frac{x^{3}}{3}$
    So, the result is: $\frac{32 x^{3}}{3}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 194 x\, dx = 194 \int x\, dx$
    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: $97 x^{2}$
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int 756\, dx = 756 x$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{2352}{x - 3}\, dx = 2352 \int \frac{1}{x - 3}\, dx$
    1. Let $u = x - 3$ .
      
      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:
      
      $\log{\left(x - 3 \right)}$
    So, the result is: $2352 \log{\left(x - 3 \right)}$
  The result is: $\frac{x^{4}}{2} + \frac{32 x^{3}}{3} + 97 x^{2} + 756 x + 2352 \log{\left(x - 3 \right)}$
Method #2
1. Rewrite the integrand:
  
  $\frac{2 \left(\left(x^{2} + 9 x\right) + 6\right)}{x - 3} \left(\left(x^{2} + 4 x\right) + 7\right) = \frac{2 x^{4} + 26 x^{3} + 98 x^{2} + 174 x + 84}{x - 3}$
2. Rewrite the integrand:
  
  $\frac{2 x^{4} + 26 x^{3} + 98 x^{2} + 174 x + 84}{x - 3} = 2 x^{3} + 32 x^{2} + 194 x + 756 + \frac{2352}{x - 3}$
3. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 2 x^{3}\, dx = 2 \int x^{3}\, dx$
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x^{3}\, dx = \frac{x^{4}}{4}$
    So, the result is: $\frac{x^{4}}{2}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 32 x^{2}\, dx = 32 \int x^{2}\, dx$
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x^{2}\, dx = \frac{x^{3}}{3}$
    So, the result is: $\frac{32 x^{3}}{3}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 194 x\, dx = 194 \int x\, dx$
    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: $97 x^{2}$
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int 756\, dx = 756 x$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{2352}{x - 3}\, dx = 2352 \int \frac{1}{x - 3}\, dx$
    1. Let $u = x - 3$ .
      
      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:
      
      $\log{\left(x - 3 \right)}$
    So, the result is: $2352 \log{\left(x - 3 \right)}$
  The result is: $\frac{x^{4}}{2} + \frac{32 x^{3}}{3} + 97 x^{2} + 756 x + 2352 \log{\left(x - 3 \right)}$
Method #3
1. Rewrite the integrand:
  
  $\frac{2 \left(\left(x^{2} + 9 x\right) + 6\right)}{x - 3} \left(\left(x^{2} + 4 x\right) + 7\right) = \frac{2 x^{4}}{x - 3} + \frac{26 x^{3}}{x - 3} + \frac{98 x^{2}}{x - 3} + \frac{174 x}{x - 3} + \frac{84}{x - 3}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{2 x^{4}}{x - 3}\, dx = 2 \int \frac{x^{4}}{x - 3}\, dx$
    1. Rewrite the integrand:
      
      $\frac{x^{4}}{x - 3} = x^{3} + 3 x^{2} + 9 x + 27 + \frac{81}{x - 3}$
    2. Integrate term-by-term:
      
      The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x^{3}\, dx = \frac{x^{4}}{4}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 3 x^{2}\, dx = 3 \int x^{2}\, dx$
      
      The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x^{2}\, dx = \frac{x^{3}}{3}$
      
      So, the result is: $x^{3}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 9 x\, dx = 9 \int x\, dx$
      
      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: $\frac{9 x^{2}}{2}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 27\, dx = 27 x$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{81}{x - 3}\, dx = 81 \int \frac{1}{x - 3}\, dx$
      
      Let $u = x - 3$ .
      
      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:
      
      $\log{\left(x - 3 \right)}$
      
      So, the result is: $81 \log{\left(x - 3 \right)}$
      
      The result is: $\frac{x^{4}}{4} + x^{3} + \frac{9 x^{2}}{2} + 27 x + 81 \log{\left(x - 3 \right)}$
    So, the result is: $\frac{x^{4}}{2} + 2 x^{3} + 9 x^{2} + 54 x + 162 \log{\left(x - 3 \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{26 x^{3}}{x - 3}\, dx = 26 \int \frac{x^{3}}{x - 3}\, dx$
    1. Rewrite the integrand:
      
      $\frac{x^{3}}{x - 3} = x^{2} + 3 x + 9 + \frac{27}{x - 3}$
    2. Integrate term-by-term:
      
      The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x^{2}\, dx = \frac{x^{3}}{3}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 3 x\, dx = 3 \int x\, dx$
      
      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: $\frac{3 x^{2}}{2}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 9\, dx = 9 x$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{27}{x - 3}\, dx = 27 \int \frac{1}{x - 3}\, dx$
      
      Let $u = x - 3$ .
      
      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:
      
      $\log{\left(x - 3 \right)}$
      
      So, the result is: $27 \log{\left(x - 3 \right)}$
      
      The result is: $\frac{x^{3}}{3} + \frac{3 x^{2}}{2} + 9 x + 27 \log{\left(x - 3 \right)}$
    So, the result is: $\frac{26 x^{3}}{3} + 39 x^{2} + 234 x + 702 \log{\left(x - 3 \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{98 x^{2}}{x - 3}\, dx = 98 \int \frac{x^{2}}{x - 3}\, dx$
    1. Rewrite the integrand:
      
      $\frac{x^{2}}{x - 3} = x + 3 + \frac{9}{x - 3}$
    2. Integrate term-by-term:
      
      The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x\, dx = \frac{x^{2}}{2}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 3\, dx = 3 x$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{9}{x - 3}\, dx = 9 \int \frac{1}{x - 3}\, dx$
      
      Let $u = x - 3$ .
      
      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:
      
      $\log{\left(x - 3 \right)}$
      
      So, the result is: $9 \log{\left(x - 3 \right)}$
      
      The result is: $\frac{x^{2}}{2} + 3 x + 9 \log{\left(x - 3 \right)}$
    So, the result is: $49 x^{2} + 294 x + 882 \log{\left(x - 3 \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{174 x}{x - 3}\, dx = 174 \int \frac{x}{x - 3}\, dx$
    1. Rewrite the integrand:
      
      $\frac{x}{x - 3} = 1 + \frac{3}{x - 3}$
    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:
      
      $\int \frac{3}{x - 3}\, dx = 3 \int \frac{1}{x - 3}\, dx$
      
      Let $u = x - 3$ .
      
      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:
      
      $\log{\left(x - 3 \right)}$
      
      So, the result is: $3 \log{\left(x - 3 \right)}$
      
      The result is: $x + 3 \log{\left(x - 3 \right)}$
    So, the result is: $174 x + 522 \log{\left(x - 3 \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{84}{x - 3}\, dx = 84 \int \frac{1}{x - 3}\, dx$
    1. Let $u = x - 3$ .
      
      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:
      
      $\log{\left(x - 3 \right)}$
    So, the result is: $84 \log{\left(x - 3 \right)}$
  The result is: $\frac{x^{4}}{2} + \frac{32 x^{3}}{3} + 97 x^{2} + 756 x + 2268 \log{\left(x - 3 \right)} + 84 \log{\left(x - 3 \right)}$
Add the constant of integration:

$\frac{x^{4}}{2} + \frac{32 x^{3}}{3} + 97 x^{2} + 756 x + 2352 \log{\left(x - 3 \right)}+ \mathrm{constant}$

The answer is:

$\frac{x^{4}}{2} + \frac{32 x^{3}}{3} + 97 x^{2} + 756 x + 2352 \log{\left(x - 3 \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                                      
 |                                                                                       
 |   / 2          \                          4                                          3
 | 2*\x  + 9*x + 6/ / 2          \          x        2                              32*x 
 | ----------------*\x  + 4*x + 7/ dx = C + -- + 97*x  + 756*x + 2352*log(-3 + x) + -----
 |      x - 3                               2                                         3  
 |                                                                                       
/

$$\int \frac{2 \left(\left(x^{2} + 9 x\right) + 6\right)}{x - 3} \left(\left(x^{2} + 4 x\right) + 7\right)\, dx = C + \frac{x^{4}}{2} + \frac{32 x^{3}}{3} + 97 x^{2} + 756 x + 2352 \log{\left(x - 3 \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

5185/6 - 2352*log(3) + 2352*log(2)

$$- 2352 \log{\left(3 \right)} + \frac{5185}{6} + 2352 \log{\left(2 \right)}$$

5185/6 - 2352*log(3) + 2352*log(2)

$$- 2352 \log{\left(3 \right)} + \frac{5185}{6} + 2352 \log{\left(2 \right)}$$

5185/6 - 2352*log(3) + 2352*log(2)

Numerical answer [src]

-89.487267603736

-89.487267603736

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

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How to use it?

Identical expressions

Similar expressions

Integral of 2(x^2+9x+6)/(x−3)(x^2+4x+7) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3