Mister Exam

Other calculators

Solving integrals/ dx/x*(1+ln(x))^5

Integral of dx/x*(1+ln(x))^5 dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1                 
  /                 
 |                  
 |              5   
 |  (1 + log(x))    
 |  ------------- dx
 |        x         
 |                  
/                   
0
∫01(log⁡(x)+1)5x dx\int\limits_{0}^{1} \frac{\left(\log{\left(x \right)} + 1\right)^{5}}{x}\, dx 
Integral((1 + log(x))^5/x, (x, 0, 1))
Detail solution

  1. There are multiple ways to do this integral.

    Method #1

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

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

      ∫(−log⁡(1u)5+5log⁡(1u)4+10log⁡(1u)3+10log⁡(1u)2+5log⁡(1u)+1u) du\int \left(- \frac{\log{\left(\frac{1}{u} \right)}^{5} + 5 \log{\left(\frac{1}{u} \right)}^{4} + 10 \log{\left(\frac{1}{u} \right)}^{3} + 10 \log{\left(\frac{1}{u} \right)}^{2} + 5 \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)5+5log⁡(1u)4+10log⁡(1u)3+10log⁡(1u)2+5log⁡(1u)+1u du=−∫log⁡(1u)5+5log⁡(1u)4+10log⁡(1u)3+10log⁡(1u)2+5log⁡(1u)+1u du\int \frac{\log{\left(\frac{1}{u} \right)}^{5} + 5 \log{\left(\frac{1}{u} \right)}^{4} + 10 \log{\left(\frac{1}{u} \right)}^{3} + 10 \log{\left(\frac{1}{u} \right)}^{2} + 5 \log{\left(\frac{1}{u} \right)} + 1}{u}\, du = - \int \frac{\log{\left(\frac{1}{u} \right)}^{5} + 5 \log{\left(\frac{1}{u} \right)}^{4} + 10 \log{\left(\frac{1}{u} \right)}^{3} + 10 \log{\left(\frac{1}{u} \right)}^{2} + 5 \log{\left(\frac{1}{u} \right)} + 1}{u}\, du

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

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

          ∫(−log⁡(u)5+5log⁡(u)4+10log⁡(u)3+10log⁡(u)2+5log⁡(u)+1u) du\int \left(- \frac{\log{\left(u \right)}^{5} + 5 \log{\left(u \right)}^{4} + 10 \log{\left(u \right)}^{3} + 10 \log{\left(u \right)}^{2} + 5 \log{\left(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⁡(u)5+5log⁡(u)4+10log⁡(u)3+10log⁡(u)2+5log⁡(u)+1u du=−∫log⁡(u)5+5log⁡(u)4+10log⁡(u)3+10log⁡(u)2+5log⁡(u)+1u du\int \frac{\log{\left(u \right)}^{5} + 5 \log{\left(u \right)}^{4} + 10 \log{\left(u \right)}^{3} + 10 \log{\left(u \right)}^{2} + 5 \log{\left(u \right)} + 1}{u}\, du = - \int \frac{\log{\left(u \right)}^{5} + 5 \log{\left(u \right)}^{4} + 10 \log{\left(u \right)}^{3} + 10 \log{\left(u \right)}^{2} + 5 \log{\left(u \right)} + 1}{u}\, du

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

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

              ∫(u5+5u4+10u3+10u2+5u+1) du\int \left(u^{5} + 5 u^{4} + 10 u^{3} + 10 u^{2} + 5 u + 1\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:

                  ∫u5 du=u66\int u^{5}\, du = \frac{u^{6}}{6}

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

                  ∫5u4 du=5∫u4 du\int 5 u^{4}\, du = 5 \int u^{4}\, du

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

                    ∫u4 du=u55\int u^{4}\, du = \frac{u^{5}}{5}

                  So, the result is: u5u^{5}

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

                  ∫10u3 du=10∫u3 du\int 10 u^{3}\, du = 10 \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: 5u42\frac{5 u^{4}}{2}

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

                  ∫10u2 du=10∫u2 du\int 10 u^{2}\, du = 10 \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: 10u33\frac{10 u^{3}}{3}

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

                  ∫5u du=5∫u du\int 5 u\, du = 5 \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: 5u22\frac{5 u^{2}}{2}

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

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

                The result is: u66+u5+5u42+10u33+5u22+u\frac{u^{6}}{6} + u^{5} + \frac{5 u^{4}}{2} + \frac{10 u^{3}}{3} + \frac{5 u^{2}}{2} + u

              Now substitute uu back in:

              log⁡(u)66+log⁡(u)5+5log⁡(u)42+10log⁡(u)33+5log⁡(u)22+log⁡(u)\frac{\log{\left(u \right)}^{6}}{6} + \log{\left(u \right)}^{5} + \frac{5 \log{\left(u \right)}^{4}}{2} + \frac{10 \log{\left(u \right)}^{3}}{3} + \frac{5 \log{\left(u \right)}^{2}}{2} + \log{\left(u \right)}

            So, the result is: −log⁡(u)66−log⁡(u)5−5log⁡(u)42−10log⁡(u)33−5log⁡(u)22−log⁡(u)- \frac{\log{\left(u \right)}^{6}}{6} - \log{\left(u \right)}^{5} - \frac{5 \log{\left(u \right)}^{4}}{2} - \frac{10 \log{\left(u \right)}^{3}}{3} - \frac{5 \log{\left(u \right)}^{2}}{2} - \log{\left(u \right)}

          Now substitute uu back in:

          −log⁡(u)66+log⁡(u)5−5log⁡(u)42+10log⁡(u)33−5log⁡(u)22+log⁡(u)- \frac{\log{\left(u \right)}^{6}}{6} + \log{\left(u \right)}^{5} - \frac{5 \log{\left(u \right)}^{4}}{2} + \frac{10 \log{\left(u \right)}^{3}}{3} - \frac{5 \log{\left(u \right)}^{2}}{2} + \log{\left(u \right)}

        So, the result is: log⁡(u)66−log⁡(u)5+5log⁡(u)42−10log⁡(u)33+5log⁡(u)22−log⁡(u)\frac{\log{\left(u \right)}^{6}}{6} - \log{\left(u \right)}^{5} + \frac{5 \log{\left(u \right)}^{4}}{2} - \frac{10 \log{\left(u \right)}^{3}}{3} + \frac{5 \log{\left(u \right)}^{2}}{2} - \log{\left(u \right)}

      Now substitute uu back in:

      log⁡(x)66+log⁡(x)5+5log⁡(x)42+10log⁡(x)33+5log⁡(x)22+log⁡(x)\frac{\log{\left(x \right)}^{6}}{6} + \log{\left(x \right)}^{5} + \frac{5 \log{\left(x \right)}^{4}}{2} + \frac{10 \log{\left(x \right)}^{3}}{3} + \frac{5 \log{\left(x \right)}^{2}}{2} + \log{\left(x \right)}

    Method #2

    1. Rewrite the integrand:

      (log⁡(x)+1)5x=log⁡(x)5+5log⁡(x)4+10log⁡(x)3+10log⁡(x)2+5log⁡(x)+1x\frac{\left(\log{\left(x \right)} + 1\right)^{5}}{x} = \frac{\log{\left(x \right)}^{5} + 5 \log{\left(x \right)}^{4} + 10 \log{\left(x \right)}^{3} + 10 \log{\left(x \right)}^{2} + 5 \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)5+5log⁡(1u)4+10log⁡(1u)3+10log⁡(1u)2+5log⁡(1u)+1u) du\int \left(- \frac{\log{\left(\frac{1}{u} \right)}^{5} + 5 \log{\left(\frac{1}{u} \right)}^{4} + 10 \log{\left(\frac{1}{u} \right)}^{3} + 10 \log{\left(\frac{1}{u} \right)}^{2} + 5 \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)5+5log⁡(1u)4+10log⁡(1u)3+10log⁡(1u)2+5log⁡(1u)+1u du=−∫log⁡(1u)5+5log⁡(1u)4+10log⁡(1u)3+10log⁡(1u)2+5log⁡(1u)+1u du\int \frac{\log{\left(\frac{1}{u} \right)}^{5} + 5 \log{\left(\frac{1}{u} \right)}^{4} + 10 \log{\left(\frac{1}{u} \right)}^{3} + 10 \log{\left(\frac{1}{u} \right)}^{2} + 5 \log{\left(\frac{1}{u} \right)} + 1}{u}\, du = - \int \frac{\log{\left(\frac{1}{u} \right)}^{5} + 5 \log{\left(\frac{1}{u} \right)}^{4} + 10 \log{\left(\frac{1}{u} \right)}^{3} + 10 \log{\left(\frac{1}{u} \right)}^{2} + 5 \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:

          ∫(−u5−5u4−10u3−10u2−5u−1) du\int \left(- u^{5} - 5 u^{4} - 10 u^{3} - 10 u^{2} - 5 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:

              ∫(−u5) du=−∫u5 du\int \left(- u^{5}\right)\, du = - \int u^{5}\, du

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

                ∫u5 du=u66\int u^{5}\, du = \frac{u^{6}}{6}

              So, the result is: −u66- \frac{u^{6}}{6}

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

              ∫(−5u4) du=−5∫u4 du\int \left(- 5 u^{4}\right)\, du = - 5 \int u^{4}\, du

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

                ∫u4 du=u55\int u^{4}\, du = \frac{u^{5}}{5}

              So, the result is: −u5- u^{5}

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

              ∫(−10u3) du=−10∫u3 du\int \left(- 10 u^{3}\right)\, du = - 10 \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: −5u42- \frac{5 u^{4}}{2}

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

              ∫(−10u2) du=−10∫u2 du\int \left(- 10 u^{2}\right)\, du = - 10 \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: −10u33- \frac{10 u^{3}}{3}

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

              ∫(−5u) du=−5∫u du\int \left(- 5 u\right)\, du = - 5 \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: −5u22- \frac{5 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: −u66−u5−5u42−10u33−5u22−u- \frac{u^{6}}{6} - u^{5} - \frac{5 u^{4}}{2} - \frac{10 u^{3}}{3} - \frac{5 u^{2}}{2} - u

          Now substitute uu back in:

          −log⁡(1u)66−log⁡(1u)5−5log⁡(1u)42−10log⁡(1u)33−5log⁡(1u)22−log⁡(1u)- \frac{\log{\left(\frac{1}{u} \right)}^{6}}{6} - \log{\left(\frac{1}{u} \right)}^{5} - \frac{5 \log{\left(\frac{1}{u} \right)}^{4}}{2} - \frac{10 \log{\left(\frac{1}{u} \right)}^{3}}{3} - \frac{5 \log{\left(\frac{1}{u} \right)}^{2}}{2} - \log{\left(\frac{1}{u} \right)}

        So, the result is: log⁡(1u)66+log⁡(1u)5+5log⁡(1u)42+10log⁡(1u)33+5log⁡(1u)22+log⁡(1u)\frac{\log{\left(\frac{1}{u} \right)}^{6}}{6} + \log{\left(\frac{1}{u} \right)}^{5} + \frac{5 \log{\left(\frac{1}{u} \right)}^{4}}{2} + \frac{10 \log{\left(\frac{1}{u} \right)}^{3}}{3} + \frac{5 \log{\left(\frac{1}{u} \right)}^{2}}{2} + \log{\left(\frac{1}{u} \right)}

      Now substitute uu back in:

      log⁡(x)66+log⁡(x)5+5log⁡(x)42+10log⁡(x)33+5log⁡(x)22+log⁡(x)\frac{\log{\left(x \right)}^{6}}{6} + \log{\left(x \right)}^{5} + \frac{5 \log{\left(x \right)}^{4}}{2} + \frac{10 \log{\left(x \right)}^{3}}{3} + \frac{5 \log{\left(x \right)}^{2}}{2} + \log{\left(x \right)}

    Method #3

    1. Rewrite the integrand:

      (log⁡(x)+1)5x=log⁡(x)5x+5log⁡(x)4x+10log⁡(x)3x+10log⁡(x)2x+5log⁡(x)x+1x\frac{\left(\log{\left(x \right)} + 1\right)^{5}}{x} = \frac{\log{\left(x \right)}^{5}}{x} + \frac{5 \log{\left(x \right)}^{4}}{x} + \frac{10 \log{\left(x \right)}^{3}}{x} + \frac{10 \log{\left(x \right)}^{2}}{x} + \frac{5 \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)5u) du\int \left(- \frac{\log{\left(\frac{1}{u} \right)}^{5}}{u}\right)\, du

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

          ∫log⁡(1u)5u du=−∫log⁡(1u)5u du\int \frac{\log{\left(\frac{1}{u} \right)}^{5}}{u}\, du = - \int \frac{\log{\left(\frac{1}{u} \right)}^{5}}{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:

            ∫(−u5) du\int \left(- u^{5}\right)\, du

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

              ∫u5 du=−∫u5 du\int u^{5}\, du = - \int u^{5}\, du

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

                ∫u5 du=u66\int u^{5}\, du = \frac{u^{6}}{6}

              So, the result is: −u66- \frac{u^{6}}{6}

            Now substitute uu back in:

            −log⁡(1u)66- \frac{\log{\left(\frac{1}{u} \right)}^{6}}{6}

          So, the result is: log⁡(1u)66\frac{\log{\left(\frac{1}{u} \right)}^{6}}{6}

        Now substitute uu back in:

        log⁡(x)66\frac{\log{\left(x \right)}^{6}}{6}

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

        ∫5log⁡(x)4x dx=5∫log⁡(x)4x dx\int \frac{5 \log{\left(x \right)}^{4}}{x}\, dx = 5 \int \frac{\log{\left(x \right)}^{4}}{x}\, dx

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

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

          ∫(−log⁡(1u)4u) du\int \left(- \frac{\log{\left(\frac{1}{u} \right)}^{4}}{u}\right)\, du

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

            ∫log⁡(1u)4u du=−∫log⁡(1u)4u du\int \frac{\log{\left(\frac{1}{u} \right)}^{4}}{u}\, du = - \int \frac{\log{\left(\frac{1}{u} \right)}^{4}}{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:

              ∫(−u4) du\int \left(- u^{4}\right)\, du

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

                ∫u4 du=−∫u4 du\int u^{4}\, du = - \int u^{4}\, du

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

                  ∫u4 du=u55\int u^{4}\, du = \frac{u^{5}}{5}

                So, the result is: −u55- \frac{u^{5}}{5}

              Now substitute uu back in:

              −log⁡(1u)55- \frac{\log{\left(\frac{1}{u} \right)}^{5}}{5}

            So, the result is: log⁡(1u)55\frac{\log{\left(\frac{1}{u} \right)}^{5}}{5}

          Now substitute uu back in:

          log⁡(x)55\frac{\log{\left(x \right)}^{5}}{5}

        So, the result is: log⁡(x)5\log{\left(x \right)}^{5}

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

        ∫10log⁡(x)3x dx=10∫log⁡(x)3x dx\int \frac{10 \log{\left(x \right)}^{3}}{x}\, dx = 10 \int \frac{\log{\left(x \right)}^{3}}{x}\, dx

        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}

        So, the result is: 5log⁡(x)42\frac{5 \log{\left(x \right)}^{4}}{2}

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

        ∫10log⁡(x)2x dx=10∫log⁡(x)2x dx\int \frac{10 \log{\left(x \right)}^{2}}{x}\, dx = 10 \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: 10log⁡(x)33\frac{10 \log{\left(x \right)}^{3}}{3}

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

        ∫5log⁡(x)x dx=5∫log⁡(x)x dx\int \frac{5 \log{\left(x \right)}}{x}\, dx = 5 \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: 5log⁡(x)22\frac{5 \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)66+log⁡(x)5+5log⁡(x)42+10log⁡(x)33+5log⁡(x)22+log⁡(x)\frac{\log{\left(x \right)}^{6}}{6} + \log{\left(x \right)}^{5} + \frac{5 \log{\left(x \right)}^{4}}{2} + \frac{10 \log{\left(x \right)}^{3}}{3} + \frac{5 \log{\left(x \right)}^{2}}{2} + \log{\left(x \right)}

  2. Now simplify:

    (log⁡(x)5+6log⁡(x)4+15log⁡(x)3+20log⁡(x)2+15log⁡(x)+6)log⁡(x)6\frac{\left(\log{\left(x \right)}^{5} + 6 \log{\left(x \right)}^{4} + 15 \log{\left(x \right)}^{3} + 20 \log{\left(x \right)}^{2} + 15 \log{\left(x \right)} + 6\right) \log{\left(x \right)}}{6}

  3. Add the constant of integration:

    (log⁡(x)5+6log⁡(x)4+15log⁡(x)3+20log⁡(x)2+15log⁡(x)+6)log⁡(x)6+constant\frac{\left(\log{\left(x \right)}^{5} + 6 \log{\left(x \right)}^{4} + 15 \log{\left(x \right)}^{3} + 20 \log{\left(x \right)}^{2} + 15 \log{\left(x \right)} + 6\right) \log{\left(x \right)}}{6}+ \mathrm{constant}

The answer is:

(log⁡(x)5+6log⁡(x)4+15log⁡(x)3+20log⁡(x)2+15log⁡(x)+6)log⁡(x)6+constant\frac{\left(\log{\left(x \right)}^{5} + 6 \log{\left(x \right)}^{4} + 15 \log{\left(x \right)}^{3} + 20 \log{\left(x \right)}^{2} + 15 \log{\left(x \right)} + 6\right) \log{\left(x \right)}}{6}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                                                      
 |                                                                                       
 |             5                       6           2           4            3            
 | (1 + log(x))              5      log (x)   5*log (x)   5*log (x)   10*log (x)         
 | ------------- dx = C + log (x) + ------- + --------- + --------- + ---------- + log(x)
 |       x                             6          2           2           3              
 |                                                                                       
/
∫(log⁡(x)+1)5x dx=C+log⁡(x)66+log⁡(x)5+5log⁡(x)42+10log⁡(x)33+5log⁡(x)22+log⁡(x)\int \frac{\left(\log{\left(x \right)} + 1\right)^{5}}{x}\, dx = C + \frac{\log{\left(x \right)}^{6}}{6} + \log{\left(x \right)}^{5} + \frac{5 \log{\left(x \right)}^{4}}{2} + \frac{10 \log{\left(x \right)}^{3}}{3} + \frac{5 \log{\left(x \right)}^{2}}{2} + \log{\left(x \right)}
Simplify
The answer [src] 
-oo
−∞-\infty 
=
= 
-oo
−∞-\infty 
-oo
Numerical answer [src] 
-1066584803.24705
-1066584803.24705

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

    © Mister Exam – Calculator