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Similar expressions
(x⁴×dx)/(x⁵+3)

Integral of (x⁴×dx)/(x⁵-3) dx

The solution

You have entered [src]

  1          
  /          
 |           
 |     4     
 |    x      
 |  ------ dx
 |   5       
 |  x  - 3   
 |           
/            
0

$$\int\limits_{0}^{1} \frac{x^{4}}{x^{5} - 3}\, dx$$

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

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = x^{5} - 3$ .
  
  Then let $du = 5 x^{4} dx$ and substitute $\frac{du}{5}$ :
  
  $\int \frac{1}{5 u}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{1}{u}\, du = \frac{\int \frac{1}{u}\, du}{5}$
    1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
    So, the result is: $\frac{\log{\left(u \right)}}{5}$
  Now substitute $u$ back in:
  
  $\frac{\log{\left(x^{5} - 3 \right)}}{5}$
Method #2
1. Let $u = x^{5}$ .
  
  Then let $du = 5 x^{4} dx$ and substitute $du$ :
  
  $\int \frac{1}{5 u - 15}\, du$
  1. There are multiple ways to do this integral.
    Method #1
    
    Let $u = 5 u - 15$ .
    
    Then let $du = 5 du$ and substitute $\frac{du}{5}$ :
    
    $\int \frac{1}{5 u}\, du$
    
    The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{1}{u}\, du = \frac{\int \frac{1}{u}\, du}{5}$
    
    The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
    
    So, the result is: $\frac{\log{\left(u \right)}}{5}$
    
    Now substitute $u$ back in:
    
    $\frac{\log{\left(5 u - 15 \right)}}{5}$
    Method #2
    
    Rewrite the integrand:
    
    $\frac{1}{5 u - 15} = \frac{1}{5 \left(u - 3\right)}$
    
    The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{1}{5 \left(u - 3\right)}\, du = \frac{\int \frac{1}{u - 3}\, du}{5}$
    
    Let $u = u - 3$ .
    
    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:
    
    $\log{\left(u - 3 \right)}$
    
    So, the result is: $\frac{\log{\left(u - 3 \right)}}{5}$
  Now substitute $u$ back in:
  
  $\frac{\log{\left(5 x^{5} - 15 \right)}}{5}$
Now simplify:

$\frac{\log{\left(x^{5} - 3 \right)}}{5}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                           
 |                            
 |    4               / 5    \
 |   x             log\x  - 3/
 | ------ dx = C + -----------
 |  5                   5     
 | x  - 3                     
 |                            
/

$$\int \frac{x^{4}}{x^{5} - 3}\, dx = C + \frac{\log{\left(x^{5} - 3 \right)}}{5}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  log(3)   log(2)
- ------ + ------
    5        5

$$- \frac{\log{\left(3 \right)}}{5} + \frac{\log{\left(2 \right)}}{5}$$

  log(3)   log(2)
- ------ + ------
    5        5

$$- \frac{\log{\left(3 \right)}}{5} + \frac{\log{\left(2 \right)}}{5}$$

-log(3)/5 + log(2)/5

Numerical answer [src]

-0.0810930216216329

-0.0810930216216329

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

Other calculators

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Identical expressions

Similar expressions

Integral of (x⁴×dx)/(x⁵-3) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #1

Method #2