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x/sqrt(five - two x)^2
x divide by square root of (5 minus 2x) squared
x divide by square root of (five minus two x) squared
x/√(5-2x)^2
x/sqrt(5-2x)2
x/sqrt5-2x2
x/sqrt(5-2x)²
x/sqrt(5-2x) to the power of 2
x/sqrt5-2x^2
x divide by sqrt(5-2x)^2
x/sqrt(5-2x)^2dx
Similar expressions
x/sqrt(5+2x)^2

Integral of x/sqrt(5-2x)^2 dx

The solution

You have entered [src]

  1                
  /                
 |                 
 |       x         
 |  ------------ dx
 |             2   
 |    _________    
 |  \/ 5 - 2*x     
 |                 
/                  
0

$$\int\limits_{0}^{1} \frac{x}{\left(\sqrt{5 - 2 x}\right)^{2}}\, dx$$

Integral(x/(sqrt(5 - 2*x))^2, (x, 0, 1))

Detail solution

Rewrite the integrand:

$\frac{x}{\left(\sqrt{5 - 2 x}\right)^{2}} = - \frac{x}{2 x - 5}$
The integral of a constant times a function is the constant times the integral of the function:

$\int \left(- \frac{x}{2 x - 5}\right)\, dx = - \int \frac{x}{2 x - 5}\, dx$
1. Rewrite the integrand:
  
  $\frac{x}{2 x - 5} = \frac{1}{2} + \frac{5}{2 \left(2 x - 5\right)}$
2. Integrate term-by-term:
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int \frac{1}{2}\, dx = \frac{x}{2}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{5}{2 \left(2 x - 5\right)}\, dx = \frac{5 \int \frac{1}{2 x - 5}\, dx}{2}$
    1. Let $u = 2 x - 5$ .
      
      Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{1}{2 u}\, du$
      1. The integral of a constant times a function is the constant times the integral of the function:
        
        $\int \frac{1}{u}\, du = \frac{\int \frac{1}{u}\, du}{2}$
        
        The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
        
        So, the result is: $\frac{\log{\left(u \right)}}{2}$
      Now substitute $u$ back in:
      
      $\frac{\log{\left(2 x - 5 \right)}}{2}$
    So, the result is: $\frac{5 \log{\left(2 x - 5 \right)}}{4}$
  The result is: $\frac{x}{2} + \frac{5 \log{\left(2 x - 5 \right)}}{4}$
So, the result is: $- \frac{x}{2} - \frac{5 \log{\left(2 x - 5 \right)}}{4}$
Add the constant of integration:

$- \frac{x}{2} - \frac{5 \log{\left(2 x - 5 \right)}}{4}+ \mathrm{constant}$

The answer is:

$- \frac{x}{2} - \frac{5 \log{\left(2 x - 5 \right)}}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                         
 |                                          
 |      x                5*log(-5 + 2*x)   x
 | ------------ dx = C - --------------- - -
 |            2                 4          2
 |   _________                              
 | \/ 5 - 2*x                               
 |                                          
/

$$\int \frac{x}{\left(\sqrt{5 - 2 x}\right)^{2}}\, dx = C - \frac{x}{2} - \frac{5 \log{\left(2 x - 5 \right)}}{4}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  1   5*log(3)   5*log(5)
- - - -------- + --------
  2      4          4

$$- \frac{5 \log{\left(3 \right)}}{4} - \frac{1}{2} + \frac{5 \log{\left(5 \right)}}{4}$$

  1   5*log(3)   5*log(5)
- - - -------- + --------
  2      4          4

$$- \frac{5 \log{\left(3 \right)}}{4} - \frac{1}{2} + \frac{5 \log{\left(5 \right)}}{4}$$

-1/2 - 5*log(3)/4 + 5*log(5)/4

Numerical answer [src]

0.138532029707488

0.138532029707488

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

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Identical expressions

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Integral of x/sqrt(5-2x)^2 dx

Limits of integration:

The graph:

Piecewise:

The solution