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Solving integrals/ xln(x^2)

Integral of xln(x^2) dx

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The solution

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∫01xlog⁡(x2) dx\int\limits_{0}^{1} x \log{\left(x^{2} \right)}\, dx 
Integral(x*log(x^2), (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}:

      ∫ueu4 du\int \frac{u e^{u}}{4}\, du

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

        ∫ueu2 du=∫ueu du2\int \frac{u e^{u}}{2}\, du = \frac{\int u e^{u}\, du}{2}

        1. Use integration by parts:

          ∫u⁡dv⁡=u⁡v⁡−∫v⁡du⁡\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.

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

          Now evaluate the sub-integral.

        2. The integral of the exponential function is itself.

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

        So, the result is: ueu2−eu2\frac{u e^{u}}{2} - \frac{e^{u}}{2}

      Now substitute uu back in:

      x2log⁡(x2)2−x22\frac{x^{2} \log{\left(x^{2} \right)}}{2} - \frac{x^{2}}{2}

    Method #2

    1. Use integration by parts:

      ∫u⁡dv⁡=u⁡v⁡−∫v⁡du⁡\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

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

      Then du⁡(x)=2x\operatorname{du}{\left(x \right)} = \frac{2}{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 n≠−1n \neq -1:

        ∫x dx=x22\int x\, dx = \frac{x^{2}}{2}

      Now evaluate the sub-integral.

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

      ∫x dx=x22\int x\, dx = \frac{x^{2}}{2}

  2. Now simplify:

    x2(log⁡(x2)−1)2\frac{x^{2} \left(\log{\left(x^{2} \right)} - 1\right)}{2}

  3. Add the constant of integration:

    x2(log⁡(x2)−1)2+constant\frac{x^{2} \left(\log{\left(x^{2} \right)} - 1\right)}{2}+ \mathrm{constant}

The answer is:

x2(log⁡(x2)−1)2+constant\frac{x^{2} \left(\log{\left(x^{2} \right)} - 1\right)}{2}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                  
 |                     2    2    / 2\
 |      / 2\          x    x *log\x /
 | x*log\x / dx = C - -- + ----------
 |                    2        2     
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2 (x2 log⁡x2−x24)2\,\left({{x^2\,\log x}\over{2}}-{{x^2}\over{4}}\right)
Simplify
The answer [src] 
-1/2
−12-{{1}\over{2}} 
=
= 
-1/2
−12- \frac{1}{2}
Simplify
Numerical answer [src] 
-0.5
-0.5

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

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