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Integral of lnx^2 dx

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The solution

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01log(x)2dx\int\limits_{0}^{1} \log{\left(x \right)}^{2}\, dx
Integral(log(x)^2, (x, 0, 1))
Detail solution
  1. Let u=log(x)u = \log{\left(x \right)}.

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

    u2eudu\int u^{2} 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)=u2u{\left(u \right)} = u^{2} and let dv(u)=eu\operatorname{dv}{\left(u \right)} = e^{u}.

      Then du(u)=2u\operatorname{du}{\left(u \right)} = 2 u.

      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. Use integration by parts:

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

      Let u(u)=2uu{\left(u \right)} = 2 u and let dv(u)=eu\operatorname{dv}{\left(u \right)} = e^{u}.

      Then du(u)=2\operatorname{du}{\left(u \right)} = 2.

      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.

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

      2eudu=2eudu\int 2 e^{u}\, du = 2 \int e^{u}\, du

      1. The integral of the exponential function is itself.

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

      So, the result is: 2eu2 e^{u}

    Now substitute uu back in:

    xlog(x)22xlog(x)+2xx \log{\left(x \right)}^{2} - 2 x \log{\left(x \right)} + 2 x

  2. Now simplify:

    x(log(x)22log(x)+2)x \left(\log{\left(x \right)}^{2} - 2 \log{\left(x \right)} + 2\right)

  3. Add the constant of integration:

    x(log(x)22log(x)+2)+constantx \left(\log{\left(x \right)}^{2} - 2 \log{\left(x \right)} + 2\right)+ \mathrm{constant}


The answer is:

x(log(x)22log(x)+2)+constantx \left(\log{\left(x \right)}^{2} - 2 \log{\left(x \right)} + 2\right)+ \mathrm{constant}

The answer (Indefinite) [src]
  /                                             
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 | log (x) dx = C + 2*x + x*log (x) - 2*x*log(x)
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log(x)2dx=C+xlog(x)22xlog(x)+2x\int \log{\left(x \right)}^{2}\, dx = C + x \log{\left(x \right)}^{2} - 2 x \log{\left(x \right)} + 2 x
The answer [src]
2
22
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=
2
22
Numerical answer [src]
2.0
2.0

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