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Identical expressions
(∛x- two ∜x/x+ three)/x
(∛x minus 2∜x divide by x plus 3) divide by x
(∛x minus two ∜x divide by x plus three) divide by x
∛x-2∜x/x+3/x
(∛x-2∜x divide by x+3) divide by x
(∛x-2∜x/x+3)/xdx
Similar expressions
(∛x+2∜x/x+3)/x
(∛x-2∜x/x-3)/x

Solving integrals/ (∛x-2∜x/x+3)/x

Integral of (∛x-2∜x/x+3)/x dx

The solution

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  1                       
  /                       
 |                        
 |            4 ___       
 |  3 ___   2*\/ x        
 |  \/ x  - ------- + 3   
 |             x          
 |  ------------------- dx
 |           x            
 |                        
/                         
0

$$\int\limits_{0}^{1} \frac{\left(- \frac{2 \sqrt[4]{x}}{x} + \sqrt[3]{x}\right) + 3}{x}\, dx$$

Integral((x^(1/3) - 2*x^(1/4)/x + 3)/x, (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = \sqrt[3]{x}$ .
  
  Then let $du = \frac{dx}{3 x^{\frac{2}{3}}}$ and substitute $du$ :
  
  $\int \frac{3 u^{\frac{13}{4}} + 9 u^{\frac{9}{4}} - 6}{u^{\frac{13}{4}}}\, du$
  1. Rewrite the integrand:
    
    $\frac{3 u^{\frac{13}{4}} + 9 u^{\frac{9}{4}} - 6}{u^{\frac{13}{4}}} = 3 + \frac{9}{u} - \frac{6}{u^{\frac{13}{4}}}$
  2. Integrate term-by-term:
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int 3\, du = 3 u$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{9}{u}\, du = 9 \int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      So, the result is: $9 \log{\left(u \right)}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{6}{u^{\frac{13}{4}}}\right)\, du = - 6 \int \frac{1}{u^{\frac{13}{4}}}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{\frac{13}{4}}}\, du = - \frac{4}{9 u^{\frac{9}{4}}}$
      
      So, the result is: $\frac{8}{3 u^{\frac{9}{4}}}$
    The result is: $3 u + 9 \log{\left(u \right)} + \frac{8}{3 u^{\frac{9}{4}}}$
  Now substitute $u$ back in:
  
  $3 \sqrt[3]{x} + 9 \log{\left(\sqrt[3]{x} \right)} + \frac{8}{3 x^{\frac{3}{4}}}$
Method #2
1. Rewrite the integrand:
  
  $\frac{\left(- \frac{2 \sqrt[4]{x}}{x} + \sqrt[3]{x}\right) + 3}{x} = \frac{x^{\frac{13}{12}} + 3 x^{\frac{3}{4}} - 2}{x^{\frac{7}{4}}}$
2. Let $u = x^{\frac{3}{4}}$ .
  
  Then let $du = \frac{3 dx}{4 \sqrt[4]{x}}$ and substitute $\frac{du}{3}$ :
  
  $\int \frac{4 u^{\frac{13}{9}} + 12 u - 8}{3 u^{2}}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{4 u^{\frac{13}{9}} + 12 u - 8}{u^{2}}\, du = \frac{\int \frac{4 u^{\frac{13}{9}} + 12 u - 8}{u^{2}}\, du}{3}$
    1. Rewrite the integrand:
      
      $\frac{4 u^{\frac{13}{9}} + 12 u - 8}{u^{2}} = \frac{12}{u} - \frac{8}{u^{2}} + \frac{4}{u^{\frac{5}{9}}}$
    2. Integrate term-by-term:
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{12}{u}\, du = 12 \int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      So, the result is: $12 \log{\left(u \right)}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{8}{u^{2}}\right)\, du = - 8 \int \frac{1}{u^{2}}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{2}}\, du = - \frac{1}{u}$
      
      So, the result is: $\frac{8}{u}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{4}{u^{\frac{5}{9}}}\, du = 4 \int \frac{1}{u^{\frac{5}{9}}}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{\frac{5}{9}}}\, du = \frac{9 u^{\frac{4}{9}}}{4}$
      
      So, the result is: $9 u^{\frac{4}{9}}$
      
      The result is: $9 u^{\frac{4}{9}} + 12 \log{\left(u \right)} + \frac{8}{u}$
    So, the result is: $3 u^{\frac{4}{9}} + 4 \log{\left(u \right)} + \frac{8}{3 u}$
  Now substitute $u$ back in:
  
  $3 \sqrt[3]{x} + 4 \log{\left(x^{\frac{3}{4}} \right)} + \frac{8}{3 x^{\frac{3}{4}}}$
Method #3
1. Rewrite the integrand:
  
  $\frac{\left(- \frac{2 \sqrt[4]{x}}{x} + \sqrt[3]{x}\right) + 3}{x} = \frac{3}{x} + \frac{1}{x^{\frac{2}{3}}} - \frac{2}{x^{\frac{7}{4}}}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{3}{x}\, dx = 3 \int \frac{1}{x}\, dx$
    1. The integral of $\frac{1}{x}$ is $\log{\left(x \right)}$ .
    So, the result is: $3 \log{\left(x \right)}$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int \frac{1}{x^{\frac{2}{3}}}\, dx = 3 \sqrt[3]{x}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{2}{x^{\frac{7}{4}}}\right)\, dx = - 2 \int \frac{1}{x^{\frac{7}{4}}}\, dx$
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{x^{\frac{7}{4}}}\, dx = - \frac{4}{3 x^{\frac{3}{4}}}$
    So, the result is: $\frac{8}{3 x^{\frac{3}{4}}}$
  The result is: $3 \sqrt[3]{x} + 3 \log{\left(x \right)} + \frac{8}{3 x^{\frac{3}{4}}}$
Now simplify:

$3 \sqrt[3]{x} + 3 \log{\left(x \right)} + \frac{8}{3 x^{\frac{3}{4}}}$
Add the constant of integration:

$3 \sqrt[3]{x} + 3 \log{\left(x \right)} + \frac{8}{3 x^{\frac{3}{4}}}+ \mathrm{constant}$

The answer is:

$3 \sqrt[3]{x} + 3 \log{\left(x \right)} + \frac{8}{3 x^{\frac{3}{4}}}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                            
 |                                                             
 |           4 ___                                             
 | 3 ___   2*\/ x                                              
 | \/ x  - ------- + 3                                         
 |            x                   3 ___        /3 ___\     8   
 | ------------------- dx = C + 3*\/ x  + 9*log\\/ x / + ------
 |          x                                               3/4
 |                                                       3*x   
/

$$\int \frac{\left(- \frac{2 \sqrt[4]{x}}{x} + \sqrt[3]{x}\right) + 3}{x}\, dx = C + 3 \sqrt[3]{x} + 9 \log{\left(\sqrt[3]{x} \right)} + \frac{8}{3 x^{\frac{3}{4}}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

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$$-\infty$$

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$$-\infty$$

-oo

Numerical answer [src]

-605780521621320.0

-605780521621320.0

The graph

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

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Identical expressions

Similar expressions

Integral of (∛x-2∜x/x+3)/x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3