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Identical expressions
(x^ two)/(x^ two +4x+ three)
(x squared ) divide by (x squared plus 4x plus 3)
(x to the power of two) divide by (x to the power of two plus 4x plus three)
(x2)/(x2+4x+3)
x2/x2+4x+3
(x²)/(x²+4x+3)
(x to the power of 2)/(x to the power of 2+4x+3)
x^2/x^2+4x+3
(x^2) divide by (x^2+4x+3)
(x^2)/(x^2+4x+3)dx
Similar expressions
(x^2)/(x^2+4x-3)
(x^2)/(x^2-4x+3)

Solving integrals/ (x^2)/(x^2+4x+3)

Integral of (x^2)/(x^2+4x+3) dx

The solution

You have entered [src]

  1                
  /                
 |                 
 |        2        
 |       x         
 |  ------------ dx
 |   2             
 |  x  + 4*x + 3   
 |                 
/                  
0

$$\int\limits_{0}^{1} \frac{x^{2}}{\left(x^{2} + 4 x\right) + 3}\, dx$$

Integral(x^2/(x^2 + 4*x + 3), (x, 0, 1))

Detail solution

Rewrite the integrand:

$\frac{x^{2}}{\left(x^{2} + 4 x\right) + 3} = 1 - \frac{9}{2 \left(x + 3\right)} + \frac{1}{2 \left(x + 1\right)}$
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:
  
  $\int \left(- \frac{9}{2 \left(x + 3\right)}\right)\, dx = - \frac{9 \int \frac{1}{x + 3}\, dx}{2}$
  1. Let $u = x + 3$ .
    
    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:
    
    $\log{\left(x + 3 \right)}$
  So, the result is: $- \frac{9 \log{\left(x + 3 \right)}}{2}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{1}{2 \left(x + 1\right)}\, dx = \frac{\int \frac{1}{x + 1}\, dx}{2}$
  1. Let $u = x + 1$ .
    
    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:
    
    $\log{\left(x + 1 \right)}$
  So, the result is: $\frac{\log{\left(x + 1 \right)}}{2}$
The result is: $x + \frac{\log{\left(x + 1 \right)}}{2} - \frac{9 \log{\left(x + 3 \right)}}{2}$
Add the constant of integration:

$x + \frac{\log{\left(x + 1 \right)}}{2} - \frac{9 \log{\left(x + 3 \right)}}{2}+ \mathrm{constant}$

The answer is:

$x + \frac{\log{\left(x + 1 \right)}}{2} - \frac{9 \log{\left(x + 3 \right)}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                   
 |                                                    
 |       2                                            
 |      x                    log(1 + x)   9*log(3 + x)
 | ------------ dx = C + x + ---------- - ------------
 |  2                            2             2      
 | x  + 4*x + 3                                       
 |                                                    
/

$$\int \frac{x^{2}}{\left(x^{2} + 4 x\right) + 3}\, dx = C + x + \frac{\log{\left(x + 1 \right)}}{2} - \frac{9 \log{\left(x + 3 \right)}}{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

    log(2)   9*log(4)   9*log(3)
1 + ------ - -------- + --------
      2         2          2

$$- \frac{9 \log{\left(4 \right)}}{2} + \frac{\log{\left(2 \right)}}{2} + 1 + \frac{9 \log{\left(3 \right)}}{2}$$

    log(2)   9*log(4)   9*log(3)
1 + ------ - -------- + --------
      2         2          2

$$- \frac{9 \log{\left(4 \right)}}{2} + \frac{\log{\left(2 \right)}}{2} + 1 + \frac{9 \log{\left(3 \right)}}{2}$$

1 + log(2)/2 - 9*log(4)/2 + 9*log(3)/2

Numerical answer [src]

0.0520042642469585

0.0520042642469585

The graph

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

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Identical expressions

Similar expressions

Integral of (x^2)/(x^2+4x+3) dx

Limits of integration:

The graph:

Piecewise:

The solution