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three x^ two \(one -5x^ three)^3
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Similar expressions
3x^2\(1+5x^3)^3

Integral of 3x^2\(1-5x^3)^3 dx

The solution

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  1               
  /               
 |                
 |         2      
 |      3*x       
 |  ----------- dx
 |            3   
 |  /       3\    
 |  \1 - 5*x /    
 |                
/                 
0

$$\int\limits_{0}^{1} \frac{3 x^{2}}{\left(1 - 5 x^{3}\right)^{3}}\, dx$$

Integral((3*x^2)/(1 - 5*x^3)^3, (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\frac{3 x^{2}}{\left(1 - 5 x^{3}\right)^{3}} = - \frac{3 x^{2}}{\left(5 x^{3} - 1\right)^{3}}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{3 x^{2}}{\left(5 x^{3} - 1\right)^{3}}\right)\, dx = - 3 \int \frac{x^{2}}{\left(5 x^{3} - 1\right)^{3}}\, dx$
  1. Let $u = 5 x^{3} - 1$ .
    
    Then let $du = 15 x^{2} dx$ and substitute $\frac{du}{15}$ :
    
    $\int \frac{1}{15 u^{3}}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{u^{3}}\, du = \frac{\int \frac{1}{u^{3}}\, du}{15}$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{3}}\, du = - \frac{1}{2 u^{2}}$
      
      So, the result is: $- \frac{1}{30 u^{2}}$
    Now substitute $u$ back in:
    
    $- \frac{1}{30 \left(5 x^{3} - 1\right)^{2}}$
  So, the result is: $\frac{1}{10 \left(5 x^{3} - 1\right)^{2}}$
Method #2
1. Rewrite the integrand:
  
  $\frac{3 x^{2}}{\left(1 - 5 x^{3}\right)^{3}} = - \frac{3 x^{2}}{125 x^{9} - 75 x^{6} + 15 x^{3} - 1}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{3 x^{2}}{125 x^{9} - 75 x^{6} + 15 x^{3} - 1}\right)\, dx = - 3 \int \frac{x^{2}}{125 x^{9} - 75 x^{6} + 15 x^{3} - 1}\, dx$
  1. Let $u = x^{3}$ .
    
    Then let $du = 3 x^{2} dx$ and substitute $du$ :
    
    $\int \frac{1}{375 u^{3} - 225 u^{2} + 45 u - 3}\, du$
    1. Rewrite the integrand:
      
      $\frac{1}{375 u^{3} - 225 u^{2} + 45 u - 3} = \frac{1}{3 \left(5 u - 1\right)^{3}}$
    2. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{3 \left(5 u - 1\right)^{3}}\, du = \frac{\int \frac{1}{\left(5 u - 1\right)^{3}}\, du}{3}$
      
      Let $u = 5 u - 1$ .
      
      Then let $du = 5 du$ and substitute $\frac{du}{5}$ :
      
      $\int \frac{1}{5 u^{3}}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{u^{3}}\, du = \frac{\int \frac{1}{u^{3}}\, du}{5}$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{3}}\, du = - \frac{1}{2 u^{2}}$
      
      So, the result is: $- \frac{1}{10 u^{2}}$
      
      Now substitute $u$ back in:
      
      $- \frac{1}{10 \left(5 u - 1\right)^{2}}$
      
      So, the result is: $- \frac{1}{30 \left(5 u - 1\right)^{2}}$
    Now substitute $u$ back in:
    
    $- \frac{1}{30 \left(5 x^{3} - 1\right)^{2}}$
  So, the result is: $\frac{1}{10 \left(5 x^{3} - 1\right)^{2}}$
Method #3
1. Rewrite the integrand:
  
  $\frac{3 x^{2}}{\left(1 - 5 x^{3}\right)^{3}} = \frac{3 x^{2}}{- 125 x^{9} + 75 x^{6} - 15 x^{3} + 1}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{3 x^{2}}{- 125 x^{9} + 75 x^{6} - 15 x^{3} + 1}\, dx = 3 \int \frac{x^{2}}{- 125 x^{9} + 75 x^{6} - 15 x^{3} + 1}\, dx$
  1. Let $u = x^{3}$ .
    
    Then let $du = 3 x^{2} dx$ and substitute $- du$ :
    
    $\int \left(- \frac{1}{375 u^{3} - 225 u^{2} + 45 u - 3}\right)\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{375 u^{3} - 225 u^{2} + 45 u - 3}\, du = - \int \frac{1}{375 u^{3} - 225 u^{2} + 45 u - 3}\, du$
      
      Rewrite the integrand:
      
      $\frac{1}{375 u^{3} - 225 u^{2} + 45 u - 3} = \frac{1}{3 \left(5 u - 1\right)^{3}}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{3 \left(5 u - 1\right)^{3}}\, du = \frac{\int \frac{1}{\left(5 u - 1\right)^{3}}\, du}{3}$
      
      Let $u = 5 u - 1$ .
      
      Then let $du = 5 du$ and substitute $\frac{du}{5}$ :
      
      $\int \frac{1}{5 u^{3}}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{u^{3}}\, du = \frac{\int \frac{1}{u^{3}}\, du}{5}$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{3}}\, du = - \frac{1}{2 u^{2}}$
      
      So, the result is: $- \frac{1}{10 u^{2}}$
      
      Now substitute $u$ back in:
      
      $- \frac{1}{10 \left(5 u - 1\right)^{2}}$
      
      So, the result is: $- \frac{1}{30 \left(5 u - 1\right)^{2}}$
      
      So, the result is: $\frac{1}{30 \left(5 u - 1\right)^{2}}$
    Now substitute $u$ back in:
    
    $\frac{1}{30 \left(5 x^{3} - 1\right)^{2}}$
  So, the result is: $\frac{1}{10 \left(5 x^{3} - 1\right)^{2}}$
Add the constant of integration:

$\frac{1}{10 \left(5 x^{3} - 1\right)^{2}}+ \mathrm{constant}$

The answer is:

$\frac{1}{10 \left(5 x^{3} - 1\right)^{2}}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                    
 |                                     
 |        2                            
 |     3*x                     1       
 | ----------- dx = C + ---------------
 |           3                        2
 | /       3\              /        3\ 
 | \1 - 5*x /           10*\-1 + 5*x / 
 |                                     
/

$$\int \frac{3 x^{2}}{\left(1 - 5 x^{3}\right)^{3}}\, dx = C + \frac{1}{10 \left(5 x^{3} - 1\right)^{2}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

nan

$$\text{NaN}$$

nan

$$\text{NaN}$$

nan

Numerical answer [src]

-1141414.97746861

-1141414.97746861

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

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Identical expressions

Similar expressions

Integral of 3x^2\(1-5x^3)^3 dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3