Mister Exam

Other calculators

Integral of ln(x^2)/x dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1           
  /           
 |            
 |     / 2\   
 |  log\x /   
 |  ------- dx
 |     x      
 |            
/             
0             
01log(x2)xdx\int\limits_{0}^{1} \frac{\log{\left(x^{2} \right)}}{x}\, dx
Integral(log(x^2)/x, (x, 0, 1))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    1. Let u=log(x2)u = \log{\left(x^{2} \right)}.

      Then let du=2dxxdu = \frac{2 dx}{x} and substitute du2\frac{du}{2}:

      u4du\int \frac{u}{4}\, du

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

        u2du=udu2\int \frac{u}{2}\, du = \frac{\int u\, du}{2}

        1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n1n \neq -1:

          udu=u22\int u\, du = \frac{u^{2}}{2}

        So, the result is: u24\frac{u^{2}}{4}

      Now substitute uu back in:

      log(x2)24\frac{\log{\left(x^{2} \right)}^{2}}{4}

    Method #2

    1. Let u=x2u = x^{2}.

      Then let du=2xdxdu = 2 x dx and substitute du2\frac{du}{2}:

      log(u)4udu\int \frac{\log{\left(u \right)}}{4 u}\, du

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

        log(u)2udu=log(u)udu2\int \frac{\log{\left(u \right)}}{2 u}\, du = \frac{\int \frac{\log{\left(u \right)}}{u}\, du}{2}

        1. Let u=1uu = \frac{1}{u}.

          Then let du=duu2du = - \frac{du}{u^{2}} and substitute du- du:

          log(1u)udu\int \frac{\log{\left(\frac{1}{u} \right)}}{u}\, du

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

            (log(1u)u)du=log(1u)udu\int \left(- \frac{\log{\left(\frac{1}{u} \right)}}{u}\right)\, du = - \int \frac{\log{\left(\frac{1}{u} \right)}}{u}\, du

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

              Then let du=duudu = - \frac{du}{u} and substitute du- du:

              udu\int u\, du

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

                (u)du=udu\int \left(- u\right)\, du = - \int u\, du

                1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n1n \neq -1:

                  udu=u22\int u\, du = \frac{u^{2}}{2}

                So, the result is: u22- \frac{u^{2}}{2}

              Now substitute uu back in:

              log(1u)22- \frac{\log{\left(\frac{1}{u} \right)}^{2}}{2}

            So, the result is: log(1u)22\frac{\log{\left(\frac{1}{u} \right)}^{2}}{2}

          Now substitute uu back in:

          log(u)22\frac{\log{\left(u \right)}^{2}}{2}

        So, the result is: log(u)24\frac{\log{\left(u \right)}^{2}}{4}

      Now substitute uu back in:

      log(x2)24\frac{\log{\left(x^{2} \right)}^{2}}{4}

    Method #3

    1. Let u=1xu = \frac{1}{x}.

      Then let du=dxx2du = - \frac{dx}{x^{2}} and substitute du- du:

      log(1u2)udu\int \frac{\log{\left(\frac{1}{u^{2}} \right)}}{u}\, du

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

        (log(1u2)u)du=log(1u2)udu\int \left(- \frac{\log{\left(\frac{1}{u^{2}} \right)}}{u}\right)\, du = - \int \frac{\log{\left(\frac{1}{u^{2}} \right)}}{u}\, du

        1. Let u=log(1u2)u = \log{\left(\frac{1}{u^{2}} \right)}.

          Then let du=2duudu = - \frac{2 du}{u} and substitute du2- \frac{du}{2}:

          u4du\int \frac{u}{4}\, du

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

            (u2)du=udu2\int \left(- \frac{u}{2}\right)\, du = - \frac{\int u\, du}{2}

            1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n1n \neq -1:

              udu=u22\int u\, du = \frac{u^{2}}{2}

            So, the result is: u24- \frac{u^{2}}{4}

          Now substitute uu back in:

          log(1u2)24- \frac{\log{\left(\frac{1}{u^{2}} \right)}^{2}}{4}

        So, the result is: log(1u2)24\frac{\log{\left(\frac{1}{u^{2}} \right)}^{2}}{4}

      Now substitute uu back in:

      log(x2)24\frac{\log{\left(x^{2} \right)}^{2}}{4}

  2. Add the constant of integration:

    log(x2)24+constant\frac{\log{\left(x^{2} \right)}^{2}}{4}+ \mathrm{constant}


The answer is:

log(x2)24+constant\frac{\log{\left(x^{2} \right)}^{2}}{4}+ \mathrm{constant}

The answer (Indefinite) [src]
  /                         
 |                          
 |    / 2\             2/ 2\
 | log\x /          log \x /
 | ------- dx = C + --------
 |    x                4    
 |                          
/                           
(logx)2\left(\log x\right)^2
The answer [src]
-oo
-\infty
=
=
-oo
-\infty
Numerical answer [src]
-1943.92772683065
-1943.92772683065

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