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Identical expressions
dx/(2x+ one)cbrt(2x+ one)
dx divide by (2x plus 1) cubic root of (2x plus 1)
dx divide by (2x plus one) cubic root of (2x plus one)
dx/2x+1cbrt2x+1
dx divide by (2x+1)cbrt(2x+1)
Similar expressions
dx/(2x+1)cbrt(2x-1)
dx/(2x-1)cbrt(2x+1)

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

The solution

You have entered [src]

  1               
  /               
 |                
 |  3 _________   
 |  \/ 2*x + 1    
 |  ----------- dx
 |    2*x + 1     
 |                
/                 
0

$$\int\limits_{0}^{1} \frac{\sqrt[3]{2 x + 1}}{2 x + 1}\, dx$$

Integral((2*x + 1)^(1/3)/(2*x + 1), (x, 0, 1))

Detail solution

Let $u = 2 x + 1$ .

Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :

$\int \frac{1}{2 u^{\frac{2}{3}}}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{1}{u^{\frac{2}{3}}}\, du = \frac{\int \frac{1}{u^{\frac{2}{3}}}\, du}{2}$
  1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int \frac{1}{u^{\frac{2}{3}}}\, du = 3 \sqrt[3]{u}$
  So, the result is: $\frac{3 \sqrt[3]{u}}{2}$
Now substitute $u$ back in:

$\frac{3 \sqrt[3]{2 x + 1}}{2}$
Now simplify:

$\frac{3 \sqrt[3]{2 x + 1}}{2}$
Add the constant of integration:

$\frac{3 \sqrt[3]{2 x + 1}}{2}+ \mathrm{constant}$

The answer is:

$\frac{3 \sqrt[3]{2 x + 1}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

$$\int \frac{\sqrt[3]{2 x + 1}}{2 x + 1}\, dx = C + \frac{3 \sqrt[3]{2 x + 1}}{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

        3 ___
  3   3*\/ 3 
- - + -------
  2      2

$$- \frac{3}{2} + \frac{3 \sqrt[3]{3}}{2}$$

        3 ___
  3   3*\/ 3 
- - + -------
  2      2

$$- \frac{3}{2} + \frac{3 \sqrt[3]{3}}{2}$$

-3/2 + 3*3^(1/3)/2

Numerical answer [src]

0.663374355461113

0.663374355461113

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

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Identical expressions

Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution