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Solving integrals/ xdx/(x+1)(x+2)

Integral of xdx/(x+1)(x+2) dx

The solution

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

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

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

Detail solution

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

3/2 - log(2)

$$\frac{3}{2} - \log{\left(2 \right)}$$

3/2 - log(2)

$$\frac{3}{2} - \log{\left(2 \right)}$$

3/2 - log(2)

Numerical answer [src]

0.806852819440055

0.806852819440055

The graph

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of xdx/(x+1)(x+2) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3