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Solving integrals/ 36dx/(3-2x)^3

Integral of 36dx/(3-2x)^3 dx

The solution

You have entered [src]

  1                   
  /                   
 |                    
 |           1        
 |  36*1*---------- dx
 |                3   
 |       (3 - 2*x)    
 |                    
/                     
0

$$\int\limits_{0}^{1} 36 \cdot 1 \cdot \frac{1}{\left(3 - 2 x\right)^{3}}\, dx$$

Integral(36*1/(3 - 2*x)^3, (x, 0, 1))

Detail solution

The integral of a constant times a function is the constant times the integral of the function:

$\int 36 \cdot 1 \cdot \frac{1}{\left(3 - 2 x\right)^{3}}\, dx = 36 \int \frac{1}{\left(3 - 2 x\right)^{3}}\, dx$
1. There are multiple ways to do this integral.
  Method #1
  1. Rewrite the integrand:
    
    $\frac{1}{\left(3 - 2 x\right)^{3}} = - \frac{1}{\left(2 x - 3\right)^{3}}$
  2. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{1}{\left(2 x - 3\right)^{3}}\right)\, dx = - \int \frac{1}{\left(2 x - 3\right)^{3}}\, dx$
    1. Let $u = 2 x - 3$ .
      
      Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{1}{4 u^{3}}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{2 u^{3}}\, du = \frac{\int \frac{1}{u^{3}}\, du}{2}$
      
      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: $- \frac{1}{4 u^{2}}$
      
      Now substitute $u$ back in:
      
      $- \frac{1}{4 \left(2 x - 3\right)^{2}}$
    So, the result is: $\frac{1}{4 \left(2 x - 3\right)^{2}}$
  Method #2
  1. Rewrite the integrand:
    
    $\frac{1}{\left(3 - 2 x\right)^{3}} = - \frac{1}{8 x^{3} - 36 x^{2} + 54 x - 27}$
  2. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{1}{8 x^{3} - 36 x^{2} + 54 x - 27}\right)\, dx = - \int \frac{1}{8 x^{3} - 36 x^{2} + 54 x - 27}\, dx$
    1. Rewrite the integrand:
      
      $\frac{1}{8 x^{3} - 36 x^{2} + 54 x - 27} = \frac{1}{\left(2 x - 3\right)^{3}}$
    2. Let $u = 2 x - 3$ .
      
      Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{1}{4 u^{3}}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{2 u^{3}}\, du = \frac{\int \frac{1}{u^{3}}\, du}{2}$
      
      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: $- \frac{1}{4 u^{2}}$
      
      Now substitute $u$ back in:
      
      $- \frac{1}{4 \left(2 x - 3\right)^{2}}$
    So, the result is: $\frac{1}{4 \left(2 x - 3\right)^{2}}$
  Method #3
  1. Rewrite the integrand:
    
    $\frac{1}{\left(3 - 2 x\right)^{3}} = \frac{1}{- 8 x^{3} + 36 x^{2} - 54 x + 27}$
  2. Rewrite the integrand:
    
    $\frac{1}{- 8 x^{3} + 36 x^{2} - 54 x + 27} = - \frac{1}{\left(2 x - 3\right)^{3}}$
  3. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{1}{\left(2 x - 3\right)^{3}}\right)\, dx = - \int \frac{1}{\left(2 x - 3\right)^{3}}\, dx$
    1. Let $u = 2 x - 3$ .
      
      Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{1}{4 u^{3}}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{2 u^{3}}\, du = \frac{\int \frac{1}{u^{3}}\, du}{2}$
      
      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: $- \frac{1}{4 u^{2}}$
      
      Now substitute $u$ back in:
      
      $- \frac{1}{4 \left(2 x - 3\right)^{2}}$
    So, the result is: $\frac{1}{4 \left(2 x - 3\right)^{2}}$
So, the result is: $\frac{9}{\left(2 x - 3\right)^{2}}$
Add the constant of integration:

$\frac{9}{\left(2 x - 3\right)^{2}}+ \mathrm{constant}$

The answer is:

$\frac{9}{\left(2 x - 3\right)^{2}}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                    
 |                                     
 |          1                    9     
 | 36*1*---------- dx = C + -----------
 |               3                    2
 |      (3 - 2*x)           (-3 + 2*x) 
 |                                     
/

$${{9}\over{\left(3-2\,x\right)^2}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

$$8$$

Упростить

Numerical answer [src]

8.0

8.0

The graph

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

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Identical expressions

Similar expressions

Integral of 36dx/(3-2x)^3 dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3