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1/((5x-2)^(4/3))dx
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1/((5x+2)^(4/3))

Integral of 1/((5x-2)^(4/3)) dx

The solution

You have entered [src]

  1                
  /                
 |                 
 |       1         
 |  ------------ dx
 |           4/3   
 |  (5*x - 2)      
 |                 
/                  
0

$$\int\limits_{0}^{1} \frac{1}{\left(5 x - 2\right)^{\frac{4}{3}}}\, dx$$

Integral(1/((5*x - 2)^(4/3)), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\frac{1}{\left(5 x - 2\right)^{\frac{4}{3}}} = \frac{1}{5 x \sqrt[3]{5 x - 2} - 2 \sqrt[3]{5 x - 2}}$
2. Let $u = \sqrt[3]{5 x - 2}$ .
  
  Then let $du = \frac{5 dx}{3 \left(5 x - 2\right)^{\frac{2}{3}}}$ and substitute $\frac{3 du}{5}$ :
  
  $\int \frac{3}{5 u^{2}}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{1}{u^{2}}\, du = \frac{3 \int \frac{1}{u^{2}}\, du}{5}$
    1. 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: $- \frac{3}{5 u}$
  Now substitute $u$ back in:
  
  $- \frac{3}{5 \sqrt[3]{5 x - 2}}$
Method #2
1. Rewrite the integrand:
  
  $\frac{1}{\left(5 x - 2\right)^{\frac{4}{3}}} = \frac{1}{5 x \sqrt[3]{5 x - 2} - 2 \sqrt[3]{5 x - 2}}$
2. Let $u = \sqrt[3]{5 x - 2}$ .
  
  Then let $du = \frac{5 dx}{3 \left(5 x - 2\right)^{\frac{2}{3}}}$ and substitute $\frac{3 du}{5}$ :
  
  $\int \frac{3}{5 u^{2}}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{1}{u^{2}}\, du = \frac{3 \int \frac{1}{u^{2}}\, du}{5}$
    1. 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: $- \frac{3}{5 u}$
  Now substitute $u$ back in:
  
  $- \frac{3}{5 \sqrt[3]{5 x - 2}}$
Add the constant of integration:

$- \frac{3}{5 \sqrt[3]{5 x - 2}}+ \mathrm{constant}$

The answer is:

$- \frac{3}{5 \sqrt[3]{5 x - 2}}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                    
 |                                     
 |      1                      3       
 | ------------ dx = C - --------------
 |          4/3            3 __________
 | (5*x - 2)             5*\/ -2 + 5*x 
 |                                     
/

$$\int \frac{1}{\left(5 x - 2\right)^{\frac{4}{3}}}\, dx = C - \frac{3}{5 \sqrt[3]{5 x - 2}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

                              2/3
            /    2/3\   3*(-2)   
oo + oo*sign\(-5)   / - ---------
                            10

$$\infty - \frac{3 \left(-2\right)^{\frac{2}{3}}}{10} + \infty \operatorname{sign}{\left(\left(-5\right)^{\frac{2}{3}} \right)}$$

                              2/3
            /    2/3\   3*(-2)   
oo + oo*sign\(-5)   / - ---------
                            10

$$\infty - \frac{3 \left(-2\right)^{\frac{2}{3}}}{10} + \infty \operatorname{sign}{\left(\left(-5\right)^{\frac{2}{3}} \right)}$$

oo + oo*sign((-5)^(2/3)) - 3*(-2)^(2/3)/10

Numerical answer [src]

(3.89712981536781 + 1.44483025793354j)

(3.89712981536781 + 1.44483025793354j)

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

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Identical expressions

Similar expressions

Integral of 1/((5x-2)^(4/3)) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2