Integral of sqrt(2*x+1) dx

The solution

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

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

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

Detail solution

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:

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

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

  1     ___
- - + \/ 3 
  3

- \frac{1}{3} + \sqrt{3}

  1     ___
- - + \/ 3 
  3

- \frac{1}{3} + \sqrt{3}

-1/3 + sqrt(3)

Numerical answer [src]

1.39871747423554

1.39871747423554

The graph

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

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Identical expressions

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Integral of sqrt(2*x+1) dx

Limits of integration:

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Piecewise:

The solution