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x/(2x-1)^1/2

Solving integrals/ x/(2x+1)^1/2

Integral of x/(2x+1)^1/2 dx

The solution

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

$$\int\limits_{0}^{1} \frac{x}{\sqrt{2 x + 1}}\, dx$$

Integral(x/sqrt(2*x + 1), (x, 0, 1))

Detail solution

Let $u = \sqrt{2 x + 1}$ .

Then let $du = \frac{dx}{\sqrt{2 x + 1}}$ and substitute $du$ :

$\int \left(\frac{u^{2}}{2} - \frac{1}{2}\right)\, du$
1. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{u^{2}}{2}\, du = \frac{\int u^{2}\, du}{2}$
    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: $\frac{u^{3}}{6}$
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int \left(- \frac{1}{2}\right)\, du = - \frac{u}{2}$
  The result is: $\frac{u^{3}}{6} - \frac{u}{2}$
Now substitute $u$ back in:

$\frac{\left(2 x + 1\right)^{\frac{3}{2}}}{6} - \frac{\sqrt{2 x + 1}}{2}$
Now simplify:

$\frac{\left(x - 1\right) \sqrt{2 x + 1}}{3}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                                               
 |                        _________            3/2
 |      x               \/ 2*x + 1    (2*x + 1)   
 | ----------- dx = C - ----------- + ------------
 |   _________               2             6      
 | \/ 2*x + 1                                     
 |                                                
/

$$\int \frac{x}{\sqrt{2 x + 1}}\, dx = C + \frac{\left(2 x + 1\right)^{\frac{3}{2}}}{6} - \frac{\sqrt{2 x + 1}}{2}$$

Simplify

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Plot the graph f(x)

The answer [src]

1/3

$$\frac{1}{3}$$

1/3

$$\frac{1}{3}$$

1/3

Numerical answer [src]

0.333333333333333

0.333333333333333

The graph

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

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Identical expressions

Similar expressions

Integral of x/(2x+1)^1/2 dx

Limits of integration:

The graph:

Piecewise:

The solution