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Solving integrals/ (x^2+log^2(x))/x

Integral of (x^2+log^2(x))/x dx

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The solution

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

      ∫(u2+e2u) du\int \left(u^{2} + e^{2 u}\right)\, du

      1. Integrate term-by-term:

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

          ∫u2 du=u33\int u^{2}\, du = \frac{u^{3}}{3}

        1. There are multiple ways to do this integral.

          Method #1

          1. Let u=2uu = 2 u.

            Then let du=2dudu = 2 du and substitute du2\frac{du}{2}:

            ∫eu4 du\int \frac{e^{u}}{4}\, du

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

              ∫eu2 du=∫eu du2\int \frac{e^{u}}{2}\, du = \frac{\int e^{u}\, du}{2}

              1. The integral of the exponential function is itself.

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

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

            Now substitute uu back in:

            e2u2\frac{e^{2 u}}{2}

          Method #2

          1. Let u=e2uu = e^{2 u}.

            Then let du=2e2ududu = 2 e^{2 u} du and substitute du2\frac{du}{2}:

            ∫14 du\int \frac{1}{4}\, du

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

              ∫12 du=∫1 du2\int \frac{1}{2}\, du = \frac{\int 1\, du}{2}

              1. The integral of a constant is the constant times the variable of integration:

                ∫1 du=u\int 1\, du = u

              So, the result is: u2\frac{u}{2}

            Now substitute uu back in:

            e2u2\frac{e^{2 u}}{2}

        The result is: u33+e2u2\frac{u^{3}}{3} + \frac{e^{2 u}}{2}

      Now substitute uu back in:

      x22+log⁡(x)33\frac{x^{2}}{2} + \frac{\log{\left(x \right)}^{3}}{3}

    Method #2

    1. Rewrite the integrand:

      x2+log⁡(x)2x=x+log⁡(x)2x\frac{x^{2} + \log{\left(x \right)}^{2}}{x} = x + \frac{\log{\left(x \right)}^{2}}{x}

    2. Integrate term-by-term:

      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}

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

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

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

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

          ∫(−log⁡(1u)2u) du=−∫log⁡(1u)2u du\int \left(- \frac{\log{\left(\frac{1}{u} \right)}^{2}}{u}\right)\, du = - \int \frac{\log{\left(\frac{1}{u} \right)}^{2}}{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:

            ∫u2 du\int u^{2}\, du

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

              ∫(−u2) du=−∫u2 du\int \left(- u^{2}\right)\, du = - \int u^{2}\, du

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

                ∫u2 du=u33\int u^{2}\, du = \frac{u^{3}}{3}

              So, the result is: −u33- \frac{u^{3}}{3}

            Now substitute uu back in:

            −log⁡(1u)33- \frac{\log{\left(\frac{1}{u} \right)}^{3}}{3}

          So, the result is: log⁡(1u)33\frac{\log{\left(\frac{1}{u} \right)}^{3}}{3}

        Now substitute uu back in:

        log⁡(x)33\frac{\log{\left(x \right)}^{3}}{3}

      The result is: x22+log⁡(x)33\frac{x^{2}}{2} + \frac{\log{\left(x \right)}^{3}}{3}

  2. Add the constant of integration:

    x22+log⁡(x)33+constant\frac{x^{2}}{2} + \frac{\log{\left(x \right)}^{3}}{3}+ \mathrm{constant}

The answer is:

x22+log⁡(x)33+constant\frac{x^{2}}{2} + \frac{\log{\left(x \right)}^{3}}{3}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                  
 |                                   
 |  2      2              2      3   
 | x  + log (x)          x    log (x)
 | ------------ dx = C + -- + -------
 |      x                2       3   
 |                                   
/
(log⁡x)33+x22{{\left(\log x\right)^3}\over{3}}+{{x^2}\over{2}}
Simplify
The answer [src] 
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Numerical answer [src] 
28568.8797156332
28568.8797156332

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

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