Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of x^(-2)
Integral of sin(3x)
Integral of logx
Integral of x^2/(1+x)
Graphing y =:
(x+2)/(x-3)
Derivative of:
(x+2)/(x-3)
Canonical form:
(x+2)/(x-3)
Identical expressions
(x+ two)/(x- three)
(x plus 2) divide by (x minus 3)
(x plus two) divide by (x minus three)
x+2/x-3
(x+2) divide by (x-3)
(x+2)/(x-3)dx
Similar expressions
(x-2)/(x-3)
(x+2)/(x+3)

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

The solution

You have entered [src]

  6         
  /         
 |          
 |  x + 2   
 |  ----- dx
 |  x - 3   
 |          
/           
4

$$\int\limits_{4}^{6} \frac{x + 2}{x - 3}\, dx$$

Integral((x + 2)/(x - 3), (x, 4, 6))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\frac{x + 2}{x - 3} = 1 + \frac{5}{x - 3}$
2. 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:
    
    $\int \frac{5}{x - 3}\, dx = 5 \int \frac{1}{x - 3}\, dx$
    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: $5 \log{\left(x - 3 \right)}$
  The result is: $x + 5 \log{\left(x - 3 \right)}$
Method #2
1. Rewrite the integrand:
  
  $\frac{x + 2}{x - 3} = \frac{x}{x - 3} + \frac{2}{x - 3}$
2. Integrate term-by-term:
  1. Rewrite the integrand:
    
    $\frac{x}{x - 3} = 1 + \frac{3}{x - 3}$
  2. 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:
      
      $\int \frac{3}{x - 3}\, dx = 3 \int \frac{1}{x - 3}\, dx$
      
      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: $x + 3 \log{\left(x - 3 \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{2}{x - 3}\, dx = 2 \int \frac{1}{x - 3}\, dx$
    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: $2 \log{\left(x - 3 \right)}$
  The result is: $x + 3 \log{\left(x - 3 \right)} + 2 \log{\left(x - 3 \right)}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

2 + 5*log(3)

$$2 + 5 \log{\left(3 \right)}$$

2 + 5*log(3)

$$2 + 5 \log{\left(3 \right)}$$

2 + 5*log(3)

Numerical answer [src]

7.49306144334055

7.49306144334055

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-3) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2