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Derivative of:
(x+2)/(x^2-9)
Graphing y =:
(x+2)/(x^2-9)
Identical expressions
(x+ two)/(x^ two - nine)
(x plus 2) divide by (x squared minus 9)
(x plus two) divide by (x to the power of two minus nine)
(x+2)/(x2-9)
x+2/x2-9
(x+2)/(x²-9)
(x+2)/(x to the power of 2-9)
x+2/x^2-9
(x+2) divide by (x^2-9)
(x+2)/(x^2-9)dx
Similar expressions
(x+2)/(x^2+9)
(x-2)/(x^2-9)

Solving integrals/ (x+2)/(x^2-9)

Integral of (x+2)/(x^2-9) dx

The solution

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  1          
  /          
 |           
 |  x + 2    
 |  ------ dx
 |   2       
 |  x  - 9   
 |           
/            
0

$$\int\limits_{0}^{1} \frac{x + 2}{x^{2} - 9}\, dx$$

Integral((x + 2)/(x^2 - 1*9), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\frac{x + 2}{x^{2} - 9} = \frac{1}{6 \left(x + 3\right)} + \frac{5}{6 \left(x - 3\right)}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{1}{6 \left(x + 3\right)}\, dx = \frac{\int \frac{1}{x + 3}\, dx}{6}$
    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: $\frac{\log{\left(x + 3 \right)}}{6}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{5}{6 \left(x - 3\right)}\, dx = \frac{5 \int \frac{1}{x - 3}\, dx}{6}$
    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: $\frac{5 \log{\left(x - 3 \right)}}{6}$
  The result is: $\frac{5 \log{\left(x - 3 \right)}}{6} + \frac{\log{\left(x + 3 \right)}}{6}$
Method #2
1. Rewrite the integrand:
  
  $\frac{x + 2}{x^{2} - 9} = \frac{x}{x^{2} - 9} + \frac{2}{x^{2} - 9}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{x}{x^{2} - 9}\, dx = \frac{\int \frac{2 x}{x^{2} - 9}\, dx}{2}$
    1. Let $u = x^{2} - 9$ .
      
      Then let $du = 2 x dx$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{1}{2 u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(x^{2} - 9 \right)}$
    So, the result is: $\frac{\log{\left(x^{2} - 9 \right)}}{2}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{2}{x^{2} - 9}\, dx = 2 \int \frac{1}{x^{2} - 9}\, dx$
    1. Rewrite the integrand:
      
      $\frac{1}{x^{2} - 9} = \frac{- \frac{1}{x + 3} + \frac{1}{x - 3}}{6}$
    2. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{- \frac{1}{x + 3} + \frac{1}{x - 3}}{6}\, dx = \frac{\int \left(- \frac{1}{x + 3} + \frac{1}{x - 3}\right)\, dx}{6}$
      
      Integrate term-by-term:
      
      The integral of $\frac{1}{x - 3}$ is $\log{\left(x - 3 \right)}$ .
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{x + 3}\right)\, dx = - \int \frac{1}{x + 3}\, dx$
      
      The integral of $\frac{1}{x + 3}$ is $\log{\left(x + 3 \right)}$ .
      
      So, the result is: $- \log{\left(x + 3 \right)}$
      
      The result is: $\log{\left(x - 3 \right)} - \log{\left(x + 3 \right)}$
      
      So, the result is: $\frac{\log{\left(x - 3 \right)}}{6} - \frac{\log{\left(x + 3 \right)}}{6}$
    So, the result is: $\frac{\log{\left(x - 3 \right)}}{3} - \frac{\log{\left(x + 3 \right)}}{3}$
  The result is: $\frac{\log{\left(x - 3 \right)}}{3} - \frac{\log{\left(x + 3 \right)}}{3} + \frac{\log{\left(x^{2} - 9 \right)}}{2}$
Add the constant of integration:

$\frac{5 \log{\left(x - 3 \right)}}{6} + \frac{\log{\left(x + 3 \right)}}{6}+ \mathrm{constant}$

The answer is:

$\frac{5 \log{\left(x - 3 \right)}}{6} + \frac{\log{\left(x + 3 \right)}}{6}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

$${{\log \left(x+3\right)}\over{6}}+{{5\,\log \left(x-3\right)}\over{ 6}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

          log(4)   5*log(2)
-log(3) + ------ + --------
            6         6

$${{\log 4}\over{6}}-\log 3+{{5\,\log 2}\over{6}}$$

          log(4)   5*log(2)
-log(3) + ------ + --------
            6         6

$$- \log{\left(3 \right)} + \frac{\log{\left(4 \right)}}{6} + \frac{5 \log{\left(2 \right)}}{6}$$

Simplify

Numerical answer [src]

-0.28994057801484

-0.28994057801484

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 (x+2)/(x^2-9) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2