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1/(x^(1/3)-4)

Integral of 1/(x^(1/3)+4) dx

The solution

You have entered [src]

 oo             
  /             
 |              
 |      1       
 |  --------- dx
 |  3 ___       
 |  \/ x  + 4   
 |              
/               
1

$$\int\limits_{1}^{\infty} \frac{1}{\sqrt[3]{x} + 4}\, dx$$

Integral(1/(x^(1/3) + 4), (x, 1, oo))

Detail solution

Let $u = \sqrt[3]{x}$ .

Then let $du = \frac{dx}{3 x^{\frac{2}{3}}}$ and substitute $3 du$ :

$\int \frac{3 u^{2}}{u + 4}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{u^{2}}{u + 4}\, du = 3 \int \frac{u^{2}}{u + 4}\, du$
  1. Rewrite the integrand:
    
    $\frac{u^{2}}{u + 4} = u - 4 + \frac{16}{u + 4}$
  2. Integrate term-by-term:
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int \left(-4\right)\, du = - 4 u$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{16}{u + 4}\, du = 16 \int \frac{1}{u + 4}\, du$
      1. Let $u = u + 4$ .
        
        Then let $du = du$ and substitute $du$ :
        
        $\int \frac{1}{u}\, du$
        
        The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
        
        Now substitute $u$ back in:
        
        $\log{\left(u + 4 \right)}$
      So, the result is: $16 \log{\left(u + 4 \right)}$
    The result is: $\frac{u^{2}}{2} - 4 u + 16 \log{\left(u + 4 \right)}$
  So, the result is: $\frac{3 u^{2}}{2} - 12 u + 48 \log{\left(u + 4 \right)}$
Now substitute $u$ back in:

$\frac{3 x^{\frac{2}{3}}}{2} - 12 \sqrt[3]{x} + 48 \log{\left(\sqrt[3]{x} + 4 \right)}$
Add the constant of integration:

$\frac{3 x^{\frac{2}{3}}}{2} - 12 \sqrt[3]{x} + 48 \log{\left(\sqrt[3]{x} + 4 \right)}+ \mathrm{constant}$

The answer is:

$\frac{3 x^{\frac{2}{3}}}{2} - 12 \sqrt[3]{x} + 48 \log{\left(\sqrt[3]{x} + 4 \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                        
 |                                                      2/3
 |     1                 3 ___         /    3 ___\   3*x   
 | --------- dx = C - 12*\/ x  + 48*log\4 + \/ x / + ------
 | 3 ___                                               2   
 | \/ x  + 4                                               
 |                                                         
/

$$\int \frac{1}{\sqrt[3]{x} + 4}\, dx = C + \frac{3 x^{\frac{2}{3}}}{2} - 12 \sqrt[3]{x} + 48 \log{\left(\sqrt[3]{x} + 4 \right)}$$

Simplify

The answer [src]

oo

$$\infty$$

oo

$$\infty$$

oo

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

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Integral of 1/(x^(1/3)+4) dx

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The solution