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Similar expressions
/1+sqrt(2x+1)
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Integral of /1+sqrt(2x-1) dx

The solution

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  3                     
  /                     
 |                      
 |  /      _________\   
 |  \1 + \/ 2*x - 1 / dx
 |                      
/                       
1

$$\int\limits_{1}^{3} \left(\sqrt{2 x - 1} + 1\right)\, dx$$

Integral(1 + sqrt(2*x - 1), (x, 1, 3))

Detail solution

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

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

        ___
5   5*\/ 5 
- + -------
3      3

$$\frac{5}{3} + \frac{5 \sqrt{5}}{3}$$

        ___
5   5*\/ 5 
- + -------
3      3

$$\frac{5}{3} + \frac{5 \sqrt{5}}{3}$$

5/3 + 5*sqrt(5)/3

Numerical answer [src]

5.39344662916632

5.39344662916632

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

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Identical expressions

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Integral of /1+sqrt(2x-1) dx

Limits of integration:

The graph:

Piecewise:

The solution