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Integral of ln(x)/(x^2) dx

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The solution

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  1          
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 |  log(x)   
 |  ------ dx
 |     2     
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01log(x)x2dx\int\limits_{0}^{1} \frac{\log{\left(x \right)}}{x^{2}}\, dx
Integral(log(x)/x^2, (x, 0, 1))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

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

      Then let du=dxxdu = \frac{dx}{x} and substitute dudu:

      ueudu\int u e^{- u}\, du

      1. Let u=uu = - u.

        Then let du=dudu = - du and substitute dudu:

        ueudu\int u e^{u}\, du

        1. Use integration by parts:

          udv=uvvdu\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

          Let u(u)=uu{\left(u \right)} = u and let dv(u)=eu\operatorname{dv}{\left(u \right)} = e^{u}.

          Then du(u)=1\operatorname{du}{\left(u \right)} = 1.

          To find v(u)v{\left(u \right)}:

          1. The integral of the exponential function is itself.

            eudu=eu\int e^{u}\, du = e^{u}

          Now evaluate the sub-integral.

        2. The integral of the exponential function is itself.

          eudu=eu\int e^{u}\, du = e^{u}

        Now substitute uu back in:

        ueueu- u e^{- u} - e^{- u}

      Now substitute uu back in:

      log(x)x1x- \frac{\log{\left(x \right)}}{x} - \frac{1}{x}

    Method #2

    1. Use integration by parts:

      udv=uvvdu\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

      Let u(x)=log(x)u{\left(x \right)} = \log{\left(x \right)} and let dv(x)=1x2\operatorname{dv}{\left(x \right)} = \frac{1}{x^{2}}.

      Then du(x)=1x\operatorname{du}{\left(x \right)} = \frac{1}{x}.

      To find v(x)v{\left(x \right)}:

      1. The integral of xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n1n \neq -1:

        1x2dx=1x\int \frac{1}{x^{2}}\, dx = - \frac{1}{x}

      Now evaluate the sub-integral.

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

      (1x2)dx=1x2dx\int \left(- \frac{1}{x^{2}}\right)\, dx = - \int \frac{1}{x^{2}}\, dx

      1. The integral of xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n1n \neq -1:

        1x2dx=1x\int \frac{1}{x^{2}}\, dx = - \frac{1}{x}

      So, the result is: 1x\frac{1}{x}

  2. Now simplify:

    log(x)+1x- \frac{\log{\left(x \right)} + 1}{x}

  3. Add the constant of integration:

    log(x)+1x+constant- \frac{\log{\left(x \right)} + 1}{x}+ \mathrm{constant}


The answer is:

log(x)+1x+constant- \frac{\log{\left(x \right)} + 1}{x}+ \mathrm{constant}

The answer (Indefinite) [src]
  /                          
 |                           
 | log(x)          1   log(x)
 | ------ dx = C - - - ------
 |    2            x     x   
 |   x                       
 |                           
/                            
log(x)x2dx=Clog(x)x1x\int \frac{\log{\left(x \right)}}{x^{2}}\, dx = C - \frac{\log{\left(x \right)}}{x} - \frac{1}{x}
The answer [src]
-oo
-\infty
=
=
-oo
-\infty
-oo
Numerical answer [src]
-5.93814806236544e+20
-5.93814806236544e+20

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