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(x*cbrt(5x)- three)^ three
(x multiply by cubic root of (5x) minus 3) cubed
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(x*cbrt(5x)-3)3
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(x*cbrt(5x)-3)^3dx
Similar expressions
(x*cbrt(5x)+3)^3

Integral of (x*cbrt(5x)-3)^3 dx

The solution

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  1                    
  /                    
 |                     
 |                 3   
 |  /  3 _____    \    
 |  \x*\/ 5*x  - 3/  dx
 |                     
/                      
0

$$\int\limits_{0}^{1} \left(x \sqrt[3]{5 x} - 3\right)^{3}\, dx$$

Integral((x*(5*x)^(1/3) - 3)^3, (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\left(x \sqrt[3]{5 x} - 3\right)^{3} = - 9 \cdot 5^{\frac{2}{3}} x^{\frac{8}{3}} + 27 \sqrt[3]{5} x^{\frac{4}{3}} + 5 x^{4} - 27$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 9 \cdot 5^{\frac{2}{3}} x^{\frac{8}{3}}\right)\, dx = - 9 \cdot 5^{\frac{2}{3}} \int x^{\frac{8}{3}}\, dx$
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x^{\frac{8}{3}}\, dx = \frac{3 x^{\frac{11}{3}}}{11}$
    So, the result is: $- \frac{27 \cdot 5^{\frac{2}{3}} x^{\frac{11}{3}}}{11}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 27 \sqrt[3]{5} x^{\frac{4}{3}}\, dx = 27 \sqrt[3]{5} \int x^{\frac{4}{3}}\, dx$
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x^{\frac{4}{3}}\, dx = \frac{3 x^{\frac{7}{3}}}{7}$
    So, the result is: $\frac{81 \sqrt[3]{5} x^{\frac{7}{3}}}{7}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 5 x^{4}\, dx = 5 \int x^{4}\, dx$
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x^{4}\, dx = \frac{x^{5}}{5}$
    So, the result is: $x^{5}$
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int \left(-27\right)\, dx = - 27 x$
  The result is: $- \frac{27 \cdot 5^{\frac{2}{3}} x^{\frac{11}{3}}}{11} + \frac{81 \sqrt[3]{5} x^{\frac{7}{3}}}{7} + x^{5} - 27 x$
Method #2
1. Rewrite the integrand:
  
  $\left(x \sqrt[3]{5 x} - 3\right)^{3} = - 9 \cdot 5^{\frac{2}{3}} x^{\frac{8}{3}} + 5 x^{4} + 27 x \sqrt[3]{5} \sqrt[3]{x} - 27$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 9 \cdot 5^{\frac{2}{3}} x^{\frac{8}{3}}\right)\, dx = - 9 \cdot 5^{\frac{2}{3}} \int x^{\frac{8}{3}}\, dx$
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x^{\frac{8}{3}}\, dx = \frac{3 x^{\frac{11}{3}}}{11}$
    So, the result is: $- \frac{27 \cdot 5^{\frac{2}{3}} x^{\frac{11}{3}}}{11}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 5 x^{4}\, dx = 5 \int x^{4}\, dx$
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x^{4}\, dx = \frac{x^{5}}{5}$
    So, the result is: $x^{5}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 27 x \sqrt[3]{5} \sqrt[3]{x}\, dx = 27 \int \sqrt[3]{5} \sqrt[3]{x} x\, dx$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \sqrt[3]{5} \sqrt[3]{x} x\, dx = \sqrt[3]{5} \int \sqrt[3]{x} x\, dx$
      
      Let $u = \sqrt[3]{x}$ .
      
      Then let $du = \frac{dx}{3 x^{\frac{2}{3}}}$ and substitute $3 du$ :
      
      $\int 3 u^{6}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int u^{6}\, du = 3 \int u^{6}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{6}\, du = \frac{u^{7}}{7}$
      
      So, the result is: $\frac{3 u^{7}}{7}$
      
      Now substitute $u$ back in:
      
      $\frac{3 x^{\frac{7}{3}}}{7}$
      
      So, the result is: $\frac{3 \sqrt[3]{5} x^{\frac{7}{3}}}{7}$
    So, the result is: $\frac{81 \sqrt[3]{5} x^{\frac{7}{3}}}{7}$
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int \left(-27\right)\, dx = - 27 x$
  The result is: $- \frac{27 \cdot 5^{\frac{2}{3}} x^{\frac{11}{3}}}{11} + \frac{81 \sqrt[3]{5} x^{\frac{7}{3}}}{7} + x^{5} - 27 x$
Add the constant of integration:

$- \frac{27 \cdot 5^{\frac{2}{3}} x^{\frac{11}{3}}}{11} + \frac{81 \sqrt[3]{5} x^{\frac{7}{3}}}{7} + x^{5} - 27 x+ \mathrm{constant}$

The answer is:

$- \frac{27 \cdot 5^{\frac{2}{3}} x^{\frac{11}{3}}}{11} + \frac{81 \sqrt[3]{5} x^{\frac{7}{3}}}{7} + x^{5} - 27 x+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                   
 |                                                                    
 |                3                          2/3  11/3      3 ___  7/3
 | /  3 _____    \            5          27*5   *x       81*\/ 5 *x   
 | \x*\/ 5*x  - 3/  dx = C + x  - 27*x - ------------- + -------------
 |                                             11              7      
/

$$\int \left(x \sqrt[3]{5 x} - 3\right)^{3}\, dx = C - \frac{27 \cdot 5^{\frac{2}{3}} x^{\frac{11}{3}}}{11} + \frac{81 \sqrt[3]{5} x^{\frac{7}{3}}}{7} + x^{5} - 27 x$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

          2/3      3 ___
      27*5      81*\/ 5 
-26 - ------- + --------
         11        7

$$-26 - \frac{27 \cdot 5^{\frac{2}{3}}}{11} + \frac{81 \sqrt[3]{5}}{7}$$

          2/3      3 ___
      27*5      81*\/ 5 
-26 - ------- + --------
         11        7

$$-26 - \frac{27 \cdot 5^{\frac{2}{3}}}{11} + \frac{81 \sqrt[3]{5}}{7}$$

-26 - 27*5^(2/3)/11 + 81*5^(1/3)/7

Numerical answer [src]

-13.3902699225103

-13.3902699225103

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

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Identical expressions

Similar expressions

Integral of (x*cbrt(5x)-3)^3 dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2