Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of e^(4*x)*dx
Integral of 1/(x^2-9)
Integral of 2*x/(-1+x)
Integral of 1/(2+3x^2)
Identical expressions
one /x*sqrt(lnx- one)
1 divide by x multiply by square root of (lnx minus 1)
one divide by x multiply by square root of (lnx minus one)
1/x*√(lnx-1)
1/xsqrt(lnx-1)
1/xsqrtlnx-1
1 divide by x*sqrt(lnx-1)
1/x*sqrt(lnx-1)dx
Similar expressions
1/x*sqrt(lnx+1)

Integral of 1/x*sqrt(lnx-1) dx

The solution

You have entered [src]

  5                  
  /                  
 |                   
 |    ____________   
 |  \/ log(x) - 1    
 |  -------------- dx
 |        x          
 |                   
/                    
E

$$\int\limits_{e}^{5} \frac{\sqrt{\log{\left(x \right)} - 1}}{x}\, dx$$

Integral(sqrt(log(x) - 1)/x, (x, E, 5))

Detail solution

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

$\frac{2 \left(\log{\left(x \right)} - 1\right)^{\frac{3}{2}}}{3}+ \mathrm{constant}$

The answer is:

$\frac{2 \left(\log{\left(x \right)} - 1\right)^{\frac{3}{2}}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                          
 |                                           
 |   ____________                         3/2
 | \/ log(x) - 1           2*(-1 + log(x))   
 | -------------- dx = C + ------------------
 |       x                         3         
 |                                           
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

               3/2
2*(-1 + log(5))   
------------------
        3

$$\frac{2 \left(-1 + \log{\left(5 \right)}\right)^{\frac{3}{2}}}{3}$$

               3/2
2*(-1 + log(5))   
------------------
        3

$$\frac{2 \left(-1 + \log{\left(5 \right)}\right)^{\frac{3}{2}}}{3}$$

2*(-1 + log(5))^(3/2)/3

Numerical answer [src]

(0.317177916876842 + 8.2241843732205e-26j)

(0.317177916876842 + 8.2241843732205e-26j)

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of 1/x*sqrt(lnx-1) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2