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(cbrt(x))-(1/(x^2-1))dx
Similar expressions
(cbrt(x))-(1/(x^2+1))
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Solving integrals/ (cbrt(x))-(1/(x^2-1))

Integral of (cbrt(x))-(1/(x^2-1)) dx

The solution

You have entered [src]

  1                      
  /                      
 |                       
 |  /3 ___       1   \   
 |  |\/ x  - 1*------| dx
 |  |           2    |   
 |  \          x  - 1/   
 |                       
/                        
0

$$\int\limits_{0}^{1} \left(\sqrt[3]{x} - 1 \cdot \frac{1}{x^{2} - 1}\right)\, dx$$

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

Detail solution

Integrate term-by-term:
1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int \sqrt[3]{x}\, dx = \frac{3 x^{\frac{4}{3}}}{4}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{1}{x^{2} - 1}\right)\, dx = - \int \frac{1}{x^{2} - 1}\, dx$
  1. Rewrite the integrand:
    
    $\frac{1}{x^{2} - 1} = \frac{- \frac{1}{x + 1} + \frac{1}{x - 1}}{2}$
  2. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{- \frac{1}{x + 1} + \frac{1}{x - 1}}{2}\, dx = \frac{\int \left(- \frac{1}{x + 1} + \frac{1}{x - 1}\right)\, dx}{2}$
    1. Integrate term-by-term:
      1. The integral of $\frac{1}{x - 1}$ is $\log{\left(x - 1 \right)}$ .
      1. The integral of a constant times a function is the constant times the integral of the function:
        
        $\int \left(- \frac{1}{x + 1}\right)\, dx = - \int \frac{1}{x + 1}\, dx$
        
        The integral of $\frac{1}{x + 1}$ is $\log{\left(x + 1 \right)}$ .
        
        So, the result is: $- \log{\left(x + 1 \right)}$
      The result is: $\log{\left(x - 1 \right)} - \log{\left(x + 1 \right)}$
    So, the result is: $\frac{\log{\left(x - 1 \right)}}{2} - \frac{\log{\left(x + 1 \right)}}{2}$
  So, the result is: $- \frac{\log{\left(x - 1 \right)}}{2} + \frac{\log{\left(x + 1 \right)}}{2}$
The result is: $\frac{3 x^{\frac{4}{3}}}{4} - \frac{\log{\left(x - 1 \right)}}{2} + \frac{\log{\left(x + 1 \right)}}{2}$
Add the constant of integration:

$\frac{3 x^{\frac{4}{3}}}{4} - \frac{\log{\left(x - 1 \right)}}{2} + \frac{\log{\left(x + 1 \right)}}{2}+ \mathrm{constant}$

The answer is:

$\frac{3 x^{\frac{4}{3}}}{4} - \frac{\log{\left(x - 1 \right)}}{2} + \frac{\log{\left(x + 1 \right)}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                             
 |                                                           4/3
 | /3 ___       1   \          log(1 + x)   log(-1 + x)   3*x   
 | |\/ x  - 1*------| dx = C + ---------- - ----------- + ------
 | |           2    |              2             2          4   
 | \          x  - 1/                                           
 |                                                              
/

$${{\log \left(x+1\right)}\over{2}}+{{3\,x^{{{4}\over{3}}}}\over{4}}- {{\log \left(x-1\right)}\over{2}}$$

Simplify

The answer [src]

     pi*I
oo + ----
      2

$${\it \%a}$$

     pi*I
oo + ----
      2

$$\infty + \frac{i \pi}{2}$$

Simplify

Numerical answer [src]

23.1420519833869

23.1420519833869

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

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Integral of (cbrt(x))-(1/(x^2-1)) dx

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Piecewise:

The solution