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(x^2+log^2(x))/xdx
Similar expressions
(x^2-log^2(x))/x

Integral of (x^2+log^2(x))/x dx

The solution

You have entered [src]

  1                
  /                
 |                 
 |   2      2      
 |  x  + log (x)   
 |  ------------ dx
 |       x         
 |                 
/                  
0

$$\int\limits_{0}^{1} \frac{x^{2} + \log{\left(x \right)}^{2}}{x}\, dx$$

Integral((x^2 + log(x)^2)/x, (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = \log{\left(x \right)}$ .
  
  Then let $du = \frac{dx}{x}$ and substitute $du$ :
  
  $\int \left(u^{2} + e^{2 u}\right)\, du$
  1. Integrate term-by-term:
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
    1. There are multiple ways to do this integral.
      
      Method #1
      
      Let $u = 2 u$ .
      
      Then let $du = 2 du$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{e^{u}}{4}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{e^{u}}{2}\, du = \frac{\int e^{u}\, du}{2}$
      
      The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
      
      So, the result is: $\frac{e^{u}}{2}$
      
      Now substitute $u$ back in:
      
      $\frac{e^{2 u}}{2}$
      
      Method #2
      
      Let $u = e^{2 u}$ .
      
      Then let $du = 2 e^{2 u} du$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{1}{4}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{2}\, du = \frac{\int 1\, du}{2}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      So, the result is: $\frac{u}{2}$
      
      Now substitute $u$ back in:
      
      $\frac{e^{2 u}}{2}$
    The result is: $\frac{u^{3}}{3} + \frac{e^{2 u}}{2}$
  Now substitute $u$ back in:
  
  $\frac{x^{2}}{2} + \frac{\log{\left(x \right)}^{3}}{3}$
Method #2
1. Rewrite the integrand:
  
  $\frac{x^{2} + \log{\left(x \right)}^{2}}{x} = x + \frac{\log{\left(x \right)}^{2}}{x}$
2. Integrate term-by-term:
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x\, dx = \frac{x^{2}}{2}$
  1. Let $u = \frac{1}{x}$ .
    
    Then let $du = - \frac{dx}{x^{2}}$ and substitute $- du$ :
    
    $\int \frac{\log{\left(\frac{1}{u} \right)}^{2}}{u}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{\log{\left(\frac{1}{u} \right)}^{2}}{u}\right)\, du = - \int \frac{\log{\left(\frac{1}{u} \right)}^{2}}{u}\, du$
      
      Let $u = \log{\left(\frac{1}{u} \right)}$ .
      
      Then let $du = - \frac{du}{u}$ and substitute $- du$ :
      
      $\int u^{2}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- u^{2}\right)\, du = - \int u^{2}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
      
      So, the result is: $- \frac{u^{3}}{3}$
      
      Now substitute $u$ back in:
      
      $- \frac{\log{\left(\frac{1}{u} \right)}^{3}}{3}$
      
      So, the result is: $\frac{\log{\left(\frac{1}{u} \right)}^{3}}{3}$
    Now substitute $u$ back in:
    
    $\frac{\log{\left(x \right)}^{3}}{3}$
  The result is: $\frac{x^{2}}{2} + \frac{\log{\left(x \right)}^{3}}{3}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                                  
 |                                   
 |  2      2              2      3   
 | x  + log (x)          x    log (x)
 | ------------ dx = C + -- + -------
 |      x                2       3   
 |                                   
/

$${{\left(\log x\right)^3}\over{3}}+{{x^2}\over{2}}$$

Simplify

The answer [src]

oo

$${\it \%a}$$

oo

$$\infty$$

Упростить

Numerical answer [src]

28568.8797156332

28568.8797156332

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

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Identical expressions

Similar expressions

Integral of (x^2+log^2(x))/x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #1

Method #2

Method #2