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Solving integrals/ (log(x)+1)^3/x

Integral of (log(x)+1)^3/x dx

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  E                 
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 |  (log(x) + 1)    
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∫1e(log⁡(x)+1)3x dx\int\limits_{1}^{e} \frac{\left(\log{\left(x \right)} + 1\right)^{3}}{x}\, dx 
Integral((log(x) + 1)^3/x, (x, 1, E))
Detail solution

  1. There are multiple ways to do this integral.

    Method #1

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

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

      ∫u3 du\int u^{3}\, du

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

        ∫u3 du=u44\int u^{3}\, du = \frac{u^{4}}{4}

      Now substitute uu back in:

      (log⁡(x)+1)44\frac{\left(\log{\left(x \right)} + 1\right)^{4}}{4}

    Method #2

    1. Rewrite the integrand:

      (log⁡(x)+1)3x=log⁡(x)3+3log⁡(x)2+3log⁡(x)+1x\frac{\left(\log{\left(x \right)} + 1\right)^{3}}{x} = \frac{\log{\left(x \right)}^{3} + 3 \log{\left(x \right)}^{2} + 3 \log{\left(x \right)} + 1}{x}

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

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

      ∫(−log⁡(1u)3+3log⁡(1u)2+3log⁡(1u)+1u) du\int \left(- \frac{\log{\left(\frac{1}{u} \right)}^{3} + 3 \log{\left(\frac{1}{u} \right)}^{2} + 3 \log{\left(\frac{1}{u} \right)} + 1}{u}\right)\, du

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

        ∫log⁡(1u)3+3log⁡(1u)2+3log⁡(1u)+1u du=−∫log⁡(1u)3+3log⁡(1u)2+3log⁡(1u)+1u du\int \frac{\log{\left(\frac{1}{u} \right)}^{3} + 3 \log{\left(\frac{1}{u} \right)}^{2} + 3 \log{\left(\frac{1}{u} \right)} + 1}{u}\, du = - \int \frac{\log{\left(\frac{1}{u} \right)}^{3} + 3 \log{\left(\frac{1}{u} \right)}^{2} + 3 \log{\left(\frac{1}{u} \right)} + 1}{u}\, du

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

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

          ∫(−u3−3u2−3u−1) du\int \left(- u^{3} - 3 u^{2} - 3 u - 1\right)\, du

          1. Integrate term-by-term:

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

              ∫(−u3) du=−∫u3 du\int \left(- u^{3}\right)\, du = - \int u^{3}\, du

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

                ∫u3 du=u44\int u^{3}\, du = \frac{u^{4}}{4}

              So, the result is: −u44- \frac{u^{4}}{4}

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

              ∫(−3u2) du=−3∫u2 du\int \left(- 3 u^{2}\right)\, du = - 3 \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: −u3- u^{3}

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

              ∫(−3u) du=−3∫u du\int \left(- 3 u\right)\, du = - 3 \int u\, du

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

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

              So, the result is: −3u22- \frac{3 u^{2}}{2}

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

              ∫(−1) du=−u\int \left(-1\right)\, du = - u

            The result is: −u44−u3−3u22−u- \frac{u^{4}}{4} - u^{3} - \frac{3 u^{2}}{2} - u

          Now substitute uu back in:

          −log⁡(1u)44−log⁡(1u)3−3log⁡(1u)22−log⁡(1u)- \frac{\log{\left(\frac{1}{u} \right)}^{4}}{4} - \log{\left(\frac{1}{u} \right)}^{3} - \frac{3 \log{\left(\frac{1}{u} \right)}^{2}}{2} - \log{\left(\frac{1}{u} \right)}

        So, the result is: log⁡(1u)44+log⁡(1u)3+3log⁡(1u)22+log⁡(1u)\frac{\log{\left(\frac{1}{u} \right)}^{4}}{4} + \log{\left(\frac{1}{u} \right)}^{3} + \frac{3 \log{\left(\frac{1}{u} \right)}^{2}}{2} + \log{\left(\frac{1}{u} \right)}

      Now substitute uu back in:

      log⁡(x)44+log⁡(x)3+3log⁡(x)22+log⁡(x)\frac{\log{\left(x \right)}^{4}}{4} + \log{\left(x \right)}^{3} + \frac{3 \log{\left(x \right)}^{2}}{2} + \log{\left(x \right)}

    Method #3

    1. Rewrite the integrand:

      (log⁡(x)+1)3x=log⁡(x)3x+3log⁡(x)2x+3log⁡(x)x+1x\frac{\left(\log{\left(x \right)} + 1\right)^{3}}{x} = \frac{\log{\left(x \right)}^{3}}{x} + \frac{3 \log{\left(x \right)}^{2}}{x} + \frac{3 \log{\left(x \right)}}{x} + \frac{1}{x}

    2. Integrate term-by-term:

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

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

        ∫(−log⁡(1u)3u) du\int \left(- \frac{\log{\left(\frac{1}{u} \right)}^{3}}{u}\right)\, du

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

          ∫log⁡(1u)3u du=−∫log⁡(1u)3u du\int \frac{\log{\left(\frac{1}{u} \right)}^{3}}{u}\, du = - \int \frac{\log{\left(\frac{1}{u} \right)}^{3}}{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:

            ∫(−u3) du\int \left(- u^{3}\right)\, du

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

              ∫u3 du=−∫u3 du\int u^{3}\, du = - \int u^{3}\, du

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

                ∫u3 du=u44\int u^{3}\, du = \frac{u^{4}}{4}

              So, the result is: −u44- \frac{u^{4}}{4}

            Now substitute uu back in:

            −log⁡(1u)44- \frac{\log{\left(\frac{1}{u} \right)}^{4}}{4}

          So, the result is: log⁡(1u)44\frac{\log{\left(\frac{1}{u} \right)}^{4}}{4}

        Now substitute uu back in:

        log⁡(x)44\frac{\log{\left(x \right)}^{4}}{4}

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

        ∫3log⁡(x)2x dx=3∫log⁡(x)2x dx\int \frac{3 \log{\left(x \right)}^{2}}{x}\, dx = 3 \int \frac{\log{\left(x \right)}^{2}}{x}\, dx

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

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

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

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

                ∫u2 du=−∫u2 du\int u^{2}\, 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}

        So, the result is: log⁡(x)3\log{\left(x \right)}^{3}

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

        ∫3log⁡(x)x dx=3∫log⁡(x)x dx\int \frac{3 \log{\left(x \right)}}{x}\, dx = 3 \int \frac{\log{\left(x \right)}}{x}\, dx

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

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

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

              ∫(−u) du\int \left(- u\right)\, du

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

                ∫u du=−∫u du\int u\, du = - \int u\, du

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

                  ∫u du=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⁡(x)22\frac{\log{\left(x \right)}^{2}}{2}

        So, the result is: 3log⁡(x)22\frac{3 \log{\left(x \right)}^{2}}{2}

      1. The integral of 1x\frac{1}{x} is log⁡(x)\log{\left(x \right)}.

      The result is: log⁡(x)44+log⁡(x)3+3log⁡(x)22+log⁡(x)\frac{\log{\left(x \right)}^{4}}{4} + \log{\left(x \right)}^{3} + \frac{3 \log{\left(x \right)}^{2}}{2} + \log{\left(x \right)}

  2. Now simplify:

    (log⁡(x)+1)44\frac{\left(\log{\left(x \right)} + 1\right)^{4}}{4}

  3. Add the constant of integration:

    (log⁡(x)+1)44+constant\frac{\left(\log{\left(x \right)} + 1\right)^{4}}{4}+ \mathrm{constant}

The answer is:

(log⁡(x)+1)44+constant\frac{\left(\log{\left(x \right)} + 1\right)^{4}}{4}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                    
 |                                     
 |             3                      4
 | (log(x) + 1)           (log(x) + 1) 
 | ------------- dx = C + -------------
 |       x                      4      
 |                                     
/
∫(log⁡(x)+1)3x dx=C+(log⁡(x)+1)44\int \frac{\left(\log{\left(x \right)} + 1\right)^{3}}{x}\, dx = C + \frac{\left(\log{\left(x \right)} + 1\right)^{4}}{4}
Simplify
The graph
1.01.21.41.61.82.02.22.42.605
Plot the graph f(x)
The answer [src] 
15/4
154\frac{15}{4} 
=
= 
15/4
154\frac{15}{4} 
15/4
Numerical answer [src] 
3.75
3.75

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

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