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Integral of (log10(x))/(x^3) dx

Limits of integration:

from to
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The graph:

from to

Piecewise:

The solution

You have entered [src]
  1             
  /             
 |              
 |  / log(x)\   
 |  |-------|   
 |  \log(10)/   
 |  --------- dx
 |       3      
 |      x       
 |              
/               
0               
$$\int\limits_{0}^{1} \frac{\frac{1}{\log{\left(10 \right)}} \log{\left(x \right)}}{x^{3}}\, dx$$
Integral((log(x)/log(10))/x^3, (x, 0, 1))
Detail solution
  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. 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:

      So, the result is:

    Now substitute back in:

  2. Now simplify:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                  
 |                       1     log(x)
 | / log(x)\          - ---- - ------
 | |-------|               2       2 
 | \log(10)/            4*x     2*x  
 | --------- dx = C + ---------------
 |      3                 log(10)    
 |     x                             
 |                                   
/                                    
$$\int \frac{\frac{1}{\log{\left(10 \right)}} \log{\left(x \right)}}{x^{3}}\, dx = C + \frac{- \frac{\log{\left(x \right)}}{2 x^{2}} - \frac{1}{4 x^{2}}}{\log{\left(10 \right)}}$$
The answer [src]
-oo
$$-\infty$$
=
=
-oo
$$-\infty$$
-oo
Numerical answer [src]
-1.7298374092956e+39
-1.7298374092956e+39

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