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Integral of lnx/x^3 dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1          
  /          
 |           
 |  log(x)   
 |  ------ dx
 |     3     
 |    x      
 |           
/            
0            
$$\int\limits_{0}^{1} \frac{\log{\left(x \right)}}{x^{3}}\, dx$$
Integral(log(x)/x^3, (x, 0, 1))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    1. Let .

      Then let and substitute :

      1. Use integration by parts:

        Let and let .

        Then .

        To find :

        1. Let .

          Then let and substitute :

          1. The integral of a constant times a function is the constant times the integral of the function:

            1. The integral of the exponential function is itself.

            So, the result is:

          Now substitute back in:

        Now evaluate the sub-integral.

      2. The integral of a constant times a function is the constant times the integral of the function:

        1. Let .

          Then let and substitute :

          1. The integral of a constant times a function is the constant times the integral of the function:

            1. The integral of the exponential function is itself.

            So, the result is:

          Now substitute back in:

        So, the result is:

      Now substitute back in:

    Method #2

    1. Use integration by parts:

      Let and let .

      Then .

      To find :

      1. The integral of is when :

      Now evaluate the sub-integral.

    2. The integral of a constant times a function is the constant times the integral of the function:

      1. The integral of is when :

      So, the result is:

  2. Now simplify:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                             
 |                              
 | log(x)           1     log(x)
 | ------ dx = C - ---- - ------
 |    3               2       2 
 |   x             4*x     2*x  
 |                              
/                               
$$\int \frac{\log{\left(x \right)}}{x^{3}}\, dx = C - \frac{\log{\left(x \right)}}{2 x^{2}} - \frac{1}{4 x^{2}}$$
The answer [src]
-oo
$$-\infty$$
=
=
-oo
$$-\infty$$
-oo
Numerical answer [src]
-3.98309783194748e+39
-3.98309783194748e+39

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