Mister Exam

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Identical expressions
four *(√^ three (lnx))/(3x)
4 multiply by (√ cubed (lnx)) divide by (3x)
four multiply by (√ to the power of three (lnx)) divide by (3x)
4*(√3(lnx))/(3x)
4*√3lnx/3x
4*(√³(lnx))/(3x)
4*(√ to the power of 3(lnx))/(3x)
4(√^3(lnx))/(3x)
4(√3(lnx))/(3x)
4√3lnx/3x
4√^3lnx/3x
4*(√^3(lnx)) divide by (3x)
4*(√^3(lnx))/(3x)dx

Integral of 4*(√^3(lnx))/(3x) dx

The solution

You have entered [src]

  E                 
  /                 
 |                  
 |              3   
 |      ________    
 |  4*\/ log(x)     
 |  ------------- dx
 |       3*x        
 |                  
/                   
1

$$\int\limits_{1}^{e} \frac{4 \left(\sqrt{\log{\left(x \right)}}\right)^{3}}{3 x}\, dx$$

Integral((4*(sqrt(log(x)))^3)/((3*x)), (x, 1, E))

Detail solution

Let $u = \sqrt{\log{\left(x \right)}}$ .

Then let $du = \frac{dx}{2 x \sqrt{\log{\left(x \right)}}}$ and substitute $\frac{8 du}{3}$ :

$\int \frac{8 u^{4}}{3}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int u^{4}\, du = \frac{8 \int u^{4}\, du}{3}$
  1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int u^{4}\, du = \frac{u^{5}}{5}$
  So, the result is: $\frac{8 u^{5}}{15}$
Now substitute $u$ back in:

$\frac{8 \log{\left(x \right)}^{\frac{5}{2}}}{15}$
Add the constant of integration:

$\frac{8 \log{\left(x \right)}^{\frac{5}{2}}}{15}+ \mathrm{constant}$

The answer is:

$\frac{8 \log{\left(x \right)}^{\frac{5}{2}}}{15}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                  
 |                                   
 |             3                     
 |     ________                5/2   
 | 4*\/ log(x)            8*log   (x)
 | ------------- dx = C + -----------
 |      3*x                    15    
 |                                   
/

$$\int \frac{4 \left(\sqrt{\log{\left(x \right)}}\right)^{3}}{3 x}\, dx = C + \frac{8 \log{\left(x \right)}^{\frac{5}{2}}}{15}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

8/15

$$\frac{8}{15}$$

8/15

$$\frac{8}{15}$$

8/15

Numerical answer [src]

0.533333333333333

0.533333333333333

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

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Identical expressions

Integral of 4*(√^3(lnx))/(3x) dx

Limits of integration:

The graph:

Piecewise:

The solution