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Identical expressions
x* four *x^ three /(x^ two + three *x- four)
x multiply by 4 multiply by x cubed divide by (x squared plus 3 multiply by x minus 4)
x multiply by four multiply by x to the power of three divide by (x to the power of two plus three multiply by x minus four)
x*4*x3/(x2+3*x-4)
x*4*x3/x2+3*x-4
x*4*x³/(x²+3*x-4)
x*4*x to the power of 3/(x to the power of 2+3*x-4)
x4x^3/(x^2+3x-4)
x4x3/(x2+3x-4)
x4x3/x2+3x-4
x4x^3/x^2+3x-4
x*4*x^3 divide by (x^2+3*x-4)
x*4*x^3/(x^2+3*x-4)dx
Similar expressions
x*4*x^3/(x^2-3*x-4)
x*4*x^3/(x^2+3*x+4)

Integral of x4x^3/(x^2+3*x-4) dx

The solution

You have entered [src]

  8                
  /                
 |                 
 |          3      
 |     x*4*x       
 |  ------------ dx
 |   2             
 |  x  + 3*x - 4   
 |                 
/                  
-10

$$\int\limits_{-10}^{8} \frac{x^{3} \cdot 4 x}{\left(x^{2} + 3 x\right) - 4}\, dx$$

Integral(((x*4)*x^3)/(x^2 + 3*x - 4), (x, -10, 8))

Detail solution

Rewrite the integrand:

$\frac{x^{3} \cdot 4 x}{\left(x^{2} + 3 x\right) - 4} = 4 x^{2} - 12 x + 52 - \frac{1024}{5 \left(x + 4\right)} + \frac{4}{5 \left(x - 1\right)}$
Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 4 x^{2}\, dx = 4 \int x^{2}\, dx$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x^{2}\, dx = \frac{x^{3}}{3}$
  So, the result is: $\frac{4 x^{3}}{3}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- 12 x\right)\, dx = - 12 \int x\, dx$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x\, dx = \frac{x^{2}}{2}$
  So, the result is: $- 6 x^{2}$
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 52\, dx = 52 x$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{1024}{5 \left(x + 4\right)}\right)\, dx = - \frac{1024 \int \frac{1}{x + 4}\, dx}{5}$
  1. Let $u = x + 4$ .
    
    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:
    
    $\log{\left(x + 4 \right)}$
  So, the result is: $- \frac{1024 \log{\left(x + 4 \right)}}{5}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{4}{5 \left(x - 1\right)}\, dx = \frac{4 \int \frac{1}{x - 1}\, dx}{5}$
  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: $\frac{4 \log{\left(x - 1 \right)}}{5}$
The result is: $\frac{4 x^{3}}{3} - 6 x^{2} + 52 x + \frac{4 \log{\left(x - 1 \right)}}{5} - \frac{1024 \log{\left(x + 4 \right)}}{5}$
Add the constant of integration:

$\frac{4 x^{3}}{3} - 6 x^{2} + 52 x + \frac{4 \log{\left(x - 1 \right)}}{5} - \frac{1024 \log{\left(x + 4 \right)}}{5}+ \mathrm{constant}$

The answer is:

$\frac{4 x^{3}}{3} - 6 x^{2} + 52 x + \frac{4 \log{\left(x - 1 \right)}}{5} - \frac{1024 \log{\left(x + 4 \right)}}{5}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                          
 |                                                                           
 |         3                                                3                
 |    x*4*x                 2          1024*log(4 + x)   4*x    4*log(-1 + x)
 | ------------ dx = C - 6*x  + 52*x - --------------- + ---- + -------------
 |  2                                         5           3           5      
 | x  + 3*x - 4                                                              
 |                                                                           
/

$$\int \frac{x^{3} \cdot 4 x}{\left(x^{2} + 3 x\right) - 4}\, dx = C + \frac{4 x^{3}}{3} - 6 x^{2} + 52 x + \frac{4 \log{\left(x - 1 \right)}}{5} - \frac{1024 \log{\left(x + 4 \right)}}{5}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

nan

$$\text{NaN}$$

nan

$$\text{NaN}$$

nan

Numerical answer [src]

-20920.8833998075

-20920.8833998075

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of x4x^3/(x^2+3*x-4) dx

Limits of integration:

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution

Integral of x4x^3/(x^2+3*x-4) dx