Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of (x^3-17)/(x^2-4x+3)
Integral of y=x^2-x
Integral of x/(x^2-3x+3)
Integral of x√x^2+1
Graphing y =:
x^2/(4-x^2)
Limit of the function:
x^2/(4-x^2)
Identical expressions
x^ two /(four -x^ two)
x squared divide by (4 minus x squared )
x to the power of two divide by (four minus x to the power of two)
x2/(4-x2)
x2/4-x2
x²/(4-x²)
x to the power of 2/(4-x to the power of 2)
x^2/4-x^2
x^2 divide by (4-x^2)
x^2/(4-x^2)dx
Similar expressions
x^2/(4+x^2)

Solving integrals/ x^2/(4-x^2)

Integral of x^2/(4-x^2) dx

The solution

You have entered [src]

  1          
  /          
 |           
 |     2     
 |    x      
 |  ------ dx
 |       2   
 |  4 - x    
 |           
/            
0

$$\int\limits_{0}^{1} \frac{x^{2}}{4 - x^{2}}\, dx$$

Integral(x^2/(4 - x^2), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\frac{x^{2}}{4 - x^{2}} = -1 + \frac{1}{x + 2} - \frac{1}{x - 2}$
2. Integrate term-by-term:
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int \left(-1\right)\, dx = - x$
  1. Let $u = x + 2$ .
    
    Then let $du = dx$ and substitute $du$ :
    
    $\int \frac{1}{u}\, du$
    1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
    Now substitute $u$ back in:
    
    $\log{\left(x + 2 \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{1}{x - 2}\right)\, dx = - \int \frac{1}{x - 2}\, dx$
    1. 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: $- \log{\left(x - 2 \right)}$
  The result is: $- x - \log{\left(x - 2 \right)} + \log{\left(x + 2 \right)}$
Method #2
1. Rewrite the integrand:
  
  $\frac{x^{2}}{4 - x^{2}} = - \frac{x^{2}}{x^{2} - 4}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{x^{2}}{x^{2} - 4}\right)\, dx = - \int \frac{x^{2}}{x^{2} - 4}\, dx$
  1. Rewrite the integrand:
    
    $\frac{x^{2}}{x^{2} - 4} = 1 - \frac{1}{x + 2} + \frac{1}{x - 2}$
  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:
      
      $\int \left(- \frac{1}{x + 2}\right)\, dx = - \int \frac{1}{x + 2}\, dx$
      
      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: $- \log{\left(x + 2 \right)}$
    1. 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:
      
      $\log{\left(x - 2 \right)}$
    The result is: $x + \log{\left(x - 2 \right)} - \log{\left(x + 2 \right)}$
  So, the result is: $- x - \log{\left(x - 2 \right)} + \log{\left(x + 2 \right)}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                                            
 |                                             
 |    2                                        
 |   x                                         
 | ------ dx = C - x - log(-2 + x) + log(2 + x)
 |      2                                      
 | 4 - x                                       
 |                                             
/

$$\int \frac{x^{2}}{4 - x^{2}}\, dx = C - x - \log{\left(x - 2 \right)} + \log{\left(x + 2 \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

-1 + log(3)

$$-1 + \log{\left(3 \right)}$$

-1 + log(3)

$$-1 + \log{\left(3 \right)}$$

-1 + log(3)

Numerical answer [src]

0.0986122886681097

0.0986122886681097

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/(4-x^2) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2