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Integral of 1/(x+x^(1/3))
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Identical expressions
one /(x+x^(one / three))
1 divide by (x plus x to the power of (1 divide by 3))
one divide by (x plus x to the power of (one divide by three))
1/(x+x(1/3))
1/x+x1/3
1/x+x^1/3
1 divide by (x+x^(1 divide by 3))
1/(x+x^(1/3))dx
Similar expressions
1/(x-x^(1/3))

Solving integrals/ 1/(x+x^(1/3))

Integral of 1/(x+x^(1/3)) dx

The solution

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  1             
  /             
 |              
 |      1       
 |  --------- dx
 |      3 ___   
 |  x + \/ x    
 |              
/               
0

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

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

Detail solution

Let $u = \sqrt[3]{x}$ .

Then let $du = \frac{dx}{3 x^{\frac{2}{3}}}$ and substitute $3 du$ :

$\int \frac{3 u}{u^{2} + 1}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{u}{u^{2} + 1}\, du = 3 \int \frac{u}{u^{2} + 1}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{u}{u^{2} + 1}\, du = \frac{\int \frac{2 u}{u^{2} + 1}\, du}{2}$
    1. Let $u = u^{2} + 1$ .
      
      Then let $du = 2 u du$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{1}{2 u}\, du$
      1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      Now substitute $u$ back in:
      
      $\log{\left(u^{2} + 1 \right)}$
    So, the result is: $\frac{\log{\left(u^{2} + 1 \right)}}{2}$
  So, the result is: $\frac{3 \log{\left(u^{2} + 1 \right)}}{2}$
Now substitute $u$ back in:

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

3*log(2)
--------
   2

$$\frac{3 \log{\left(2 \right)}}{2}$$

3*log(2)
--------
   2

$$\frac{3 \log{\left(2 \right)}}{2}$$

3*log(2)/2

Numerical answer [src]

1.03972077083961

1.03972077083961

The graph

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

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Identical expressions

Similar expressions

Integral of 1/(x+x^(1/3)) dx

Limits of integration:

The graph:

Piecewise:

The solution