Integral of xln(x+2)(x-3) dx

The solution

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 |  x*log(x + 2)*(x - 3) dx
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$$\int\limits_{0}^{1} x \log{\left(x + 2 \right)} \left(x - 3\right)\, dx$$

Integral((x*log(x + 2))*(x - 3), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $x \log{\left(x + 2 \right)} \left(x - 3\right) = x^{2} \log{\left(x + 2 \right)} - 3 x \log{\left(x + 2 \right)}$
2. Integrate term-by-term:
  1. Use integration by parts:
    
    $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
    
    Let $u{\left(x \right)} = \log{\left(x + 2 \right)}$ and let $\operatorname{dv}{\left(x \right)} = x^{2}$ .
    
    Then $\operatorname{du}{\left(x \right)} = \frac{1}{x + 2}$ .
    
    To find $v{\left(x \right)}$ :
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x^{2}\, dx = \frac{x^{3}}{3}$
    Now evaluate the sub-integral.
  2. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{x^{3}}{3 \left(x + 2\right)}\, dx = \frac{\int \frac{x^{3}}{x + 2}\, dx}{3}$
    1. Rewrite the integrand:
      
      $\frac{x^{3}}{x + 2} = x^{2} - 2 x + 4 - \frac{8}{x + 2}$
    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:
      
      $\int \left(- 2 x\right)\, dx = - 2 \int x\, dx$
      
      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: $- x^{2}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 4\, dx = 4 x$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{8}{x + 2}\right)\, dx = - 8 \int \frac{1}{x + 2}\, dx$
      
      Let $u = x + 2$ .
      
      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:
      
      $\log{\left(x + 2 \right)}$
      
      So, the result is: $- 8 \log{\left(x + 2 \right)}$
      
      The result is: $\frac{x^{3}}{3} - x^{2} + 4 x - 8 \log{\left(x + 2 \right)}$
    So, the result is: $\frac{x^{3}}{9} - \frac{x^{2}}{3} + \frac{4 x}{3} - \frac{8 \log{\left(x + 2 \right)}}{3}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 3 x \log{\left(x + 2 \right)}\right)\, dx = - 3 \int x \log{\left(x + 2 \right)}\, dx$
    1. Use integration by parts:
      
      $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
      
      Let $u{\left(x \right)} = \log{\left(x + 2 \right)}$ and let $\operatorname{dv}{\left(x \right)} = x$ .
      
      Then $\operatorname{du}{\left(x \right)} = \frac{1}{x + 2}$ .
      
      To find $v{\left(x \right)}$ :
      
      The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x\, dx = \frac{x^{2}}{2}$
      
      Now evaluate the sub-integral.
    2. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{x^{2}}{2 \left(x + 2\right)}\, dx = \frac{\int \frac{x^{2}}{x + 2}\, dx}{2}$
      
      Rewrite the integrand:
      
      $\frac{x^{2}}{x + 2} = x - 2 + \frac{4}{x + 2}$
      
      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:
      
      $\int \left(-2\right)\, dx = - 2 x$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{4}{x + 2}\, dx = 4 \int \frac{1}{x + 2}\, dx$
      
      Let $u = x + 2$ .
      
      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:
      
      $\log{\left(x + 2 \right)}$
      
      So, the result is: $4 \log{\left(x + 2 \right)}$
      
      The result is: $\frac{x^{2}}{2} - 2 x + 4 \log{\left(x + 2 \right)}$
      
      So, the result is: $\frac{x^{2}}{4} - x + 2 \log{\left(x + 2 \right)}$
    So, the result is: $- \frac{3 x^{2} \log{\left(x + 2 \right)}}{2} + \frac{3 x^{2}}{4} - 3 x + 6 \log{\left(x + 2 \right)}$
  The result is: $\frac{x^{3} \log{\left(x + 2 \right)}}{3} - \frac{x^{3}}{9} - \frac{3 x^{2} \log{\left(x + 2 \right)}}{2} + \frac{13 x^{2}}{12} - \frac{13 x}{3} + \frac{26 \log{\left(x + 2 \right)}}{3}$
Method #2
1. Rewrite the integrand:
  
  $x \log{\left(x + 2 \right)} \left(x - 3\right) = x^{2} \log{\left(x + 2 \right)} - 3 x \log{\left(x + 2 \right)}$
2. Integrate term-by-term:
  1. Use integration by parts:
    
    $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
    
    Let $u{\left(x \right)} = \log{\left(x + 2 \right)}$ and let $\operatorname{dv}{\left(x \right)} = x^{2}$ .
    
    Then $\operatorname{du}{\left(x \right)} = \frac{1}{x + 2}$ .
    
    To find $v{\left(x \right)}$ :
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x^{2}\, dx = \frac{x^{3}}{3}$
    Now evaluate the sub-integral.
  2. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{x^{3}}{3 \left(x + 2\right)}\, dx = \frac{\int \frac{x^{3}}{x + 2}\, dx}{3}$
    1. Rewrite the integrand:
      
      $\frac{x^{3}}{x + 2} = x^{2} - 2 x + 4 - \frac{8}{x + 2}$
    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:
      
      $\int \left(- 2 x\right)\, dx = - 2 \int x\, dx$
      
      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: $- x^{2}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 4\, dx = 4 x$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{8}{x + 2}\right)\, dx = - 8 \int \frac{1}{x + 2}\, dx$
      
      Let $u = x + 2$ .
      
      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:
      
      $\log{\left(x + 2 \right)}$
      
      So, the result is: $- 8 \log{\left(x + 2 \right)}$
      
      The result is: $\frac{x^{3}}{3} - x^{2} + 4 x - 8 \log{\left(x + 2 \right)}$
    So, the result is: $\frac{x^{3}}{9} - \frac{x^{2}}{3} + \frac{4 x}{3} - \frac{8 \log{\left(x + 2 \right)}}{3}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 3 x \log{\left(x + 2 \right)}\right)\, dx = - 3 \int x \log{\left(x + 2 \right)}\, dx$
    1. Use integration by parts:
      
      $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
      
      Let $u{\left(x \right)} = \log{\left(x + 2 \right)}$ and let $\operatorname{dv}{\left(x \right)} = x$ .
      
      Then $\operatorname{du}{\left(x \right)} = \frac{1}{x + 2}$ .
      
      To find $v{\left(x \right)}$ :
      
      The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x\, dx = \frac{x^{2}}{2}$
      
      Now evaluate the sub-integral.
    2. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{x^{2}}{2 \left(x + 2\right)}\, dx = \frac{\int \frac{x^{2}}{x + 2}\, dx}{2}$
      
      Rewrite the integrand:
      
      $\frac{x^{2}}{x + 2} = x - 2 + \frac{4}{x + 2}$
      
      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:
      
      $\int \left(-2\right)\, dx = - 2 x$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{4}{x + 2}\, dx = 4 \int \frac{1}{x + 2}\, dx$
      
      Let $u = x + 2$ .
      
      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:
      
      $\log{\left(x + 2 \right)}$
      
      So, the result is: $4 \log{\left(x + 2 \right)}$
      
      The result is: $\frac{x^{2}}{2} - 2 x + 4 \log{\left(x + 2 \right)}$
      
      So, the result is: $\frac{x^{2}}{4} - x + 2 \log{\left(x + 2 \right)}$
    So, the result is: $- \frac{3 x^{2} \log{\left(x + 2 \right)}}{2} + \frac{3 x^{2}}{4} - 3 x + 6 \log{\left(x + 2 \right)}$
  The result is: $\frac{x^{3} \log{\left(x + 2 \right)}}{3} - \frac{x^{3}}{9} - \frac{3 x^{2} \log{\left(x + 2 \right)}}{2} + \frac{13 x^{2}}{12} - \frac{13 x}{3} + \frac{26 \log{\left(x + 2 \right)}}{3}$
Add the constant of integration:

$\frac{x^{3} \log{\left(x + 2 \right)}}{3} - \frac{x^{3}}{9} - \frac{3 x^{2} \log{\left(x + 2 \right)}}{2} + \frac{13 x^{2}}{12} - \frac{13 x}{3} + \frac{26 \log{\left(x + 2 \right)}}{3}+ \mathrm{constant}$

The answer is:

$\frac{x^{3} \log{\left(x + 2 \right)}}{3} - \frac{x^{3}}{9} - \frac{3 x^{2} \log{\left(x + 2 \right)}}{2} + \frac{13 x^{2}}{12} - \frac{13 x}{3} + \frac{26 \log{\left(x + 2 \right)}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                      3       2                      2               3           
 |                               13*x   x    13*x    26*log(2 + x)   3*x *log(2 + x)   x *log(2 + x)
 | x*log(x + 2)*(x - 3) dx = C - ---- - -- + ----- + ------------- - --------------- + -------------
 |                                3     9      12          3                2                3      
/

$$\int x \log{\left(x + 2 \right)} \left(x - 3\right)\, dx = C + \frac{x^{3} \log{\left(x + 2 \right)}}{3} - \frac{x^{3}}{9} - \frac{3 x^{2} \log{\left(x + 2 \right)}}{2} + \frac{13 x^{2}}{12} - \frac{13 x}{3} + \frac{26 \log{\left(x + 2 \right)}}{3}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  121   26*log(2)   15*log(3)
- --- - --------- + ---------
   36       3           2

$$- \frac{26 \log{\left(2 \right)}}{3} - \frac{121}{36} + \frac{15 \log{\left(3 \right)}}{2}$$

  121   26*log(2)   15*log(3)
- --- - --------- + ---------
   36       3           2

$$- \frac{26 \log{\left(2 \right)}}{3} - \frac{121}{36} + \frac{15 \log{\left(3 \right)}}{2}$$

-121/36 - 26*log(2)/3 + 15*log(3)/2

Numerical answer [src]

-1.12879451095315

-1.12879451095315

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

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Integral of xln(x+2)(x-3) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2