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x/sqrt(five - four *x)
x divide by square root of (5 minus 4 multiply by x)
x divide by square root of (five minus four multiply by x)
x/√(5-4*x)
x/sqrt(5-4x)
x/sqrt5-4x
x divide by sqrt(5-4*x)
x/sqrt(5-4*x)dx
Similar expressions
x/sqrt(5+4*x)
x/sqrt(5-4x)

Integral of x/sqrt(5-4*x) dx

The solution

You have entered [src]

  1               
  /               
 |                
 |       x        
 |  ----------- dx
 |    _________   
 |  \/ 5 - 4*x    
 |                
/                 
0

$$\int\limits_{0}^{1} \frac{x}{\sqrt{5 - 4 x}}\, dx$$

Integral(x/sqrt(5 - 4*x), (x, 0, 1))

Detail solution

Let $u = \sqrt{5 - 4 x}$ .

Then let $du = - \frac{2 dx}{\sqrt{5 - 4 x}}$ and substitute $du$ :

$\int \left(\frac{u^{2}}{8} - \frac{5}{8}\right)\, du$
1. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{u^{2}}{8}\, du = \frac{\int u^{2}\, du}{8}$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
    So, the result is: $\frac{u^{3}}{24}$
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int \left(- \frac{5}{8}\right)\, du = - \frac{5 u}{8}$
  The result is: $\frac{u^{3}}{24} - \frac{5 u}{8}$
Now substitute $u$ back in:

$\frac{\left(5 - 4 x\right)^{\frac{3}{2}}}{24} - \frac{5 \sqrt{5 - 4 x}}{8}$
Now simplify:

$- \frac{\sqrt{5 - 4 x} \left(2 x + 5\right)}{12}$
Add the constant of integration:

$- \frac{\sqrt{5 - 4 x} \left(2 x + 5\right)}{12}+ \mathrm{constant}$

The answer is:

$- \frac{\sqrt{5 - 4 x} \left(2 x + 5\right)}{12}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                 
 |                          _________            3/2
 |      x               5*\/ 5 - 4*x    (5 - 4*x)   
 | ----------- dx = C - ------------- + ------------
 |   _________                8              24     
 | \/ 5 - 4*x                                       
 |                                                  
/

$$\int \frac{x}{\sqrt{5 - 4 x}}\, dx = C + \frac{\left(5 - 4 x\right)^{\frac{3}{2}}}{24} - \frac{5 \sqrt{5 - 4 x}}{8}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

           ___
  7    5*\/ 5 
- -- + -------
  12      12

$$- \frac{7}{12} + \frac{5 \sqrt{5}}{12}$$

           ___
  7    5*\/ 5 
- -- + -------
  12      12

$$- \frac{7}{12} + \frac{5 \sqrt{5}}{12}$$

-7/12 + 5*sqrt(5)/12

Numerical answer [src]

0.348361657291579

0.348361657291579

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(5-4*x) dx

Limits of integration:

The graph:

Piecewise:

The solution