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

Limits of integration:

from to
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The graph:

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Piecewise:

The solution

You have entered [src]
  1                 
  /                 
 |                  
 |              5   
 |  (1 + log(x))    
 |  ------------- dx
 |        x         
 |                  
/                   
0                   
$$\int\limits_{0}^{1} \frac{\left(\log{\left(x \right)} + 1\right)^{5}}{x}\, dx$$
Integral((1 + log(x))^5/x, (x, 0, 1))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    1. Let .

      Then let and substitute :

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

        1. Let .

          Then let and substitute :

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

            1. Let .

              Then let and substitute :

              1. Integrate term-by-term:

                1. The integral of is when :

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

                  1. The integral of is when :

                  So, the result is:

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

                  1. The integral of is when :

                  So, the result is:

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

                  1. The integral of is when :

                  So, the result is:

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

                  1. The integral of is when :

                  So, the result is:

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

                The result is:

              Now substitute back in:

            So, the result is:

          Now substitute back in:

        So, the result is:

      Now substitute back in:

    Method #2

    1. Rewrite the integrand:

    2. Let .

      Then let and substitute :

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

        1. Let .

          Then let and substitute :

          1. Integrate term-by-term:

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

              1. The integral of is when :

              So, the result is:

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

              1. The integral of is when :

              So, the result is:

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

              1. The integral of is when :

              So, the result is:

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

              1. The integral of is when :

              So, the result is:

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

              1. The integral of is when :

              So, the result is:

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

            The result is:

          Now substitute back in:

        So, the result is:

      Now substitute back in:

    Method #3

    1. Rewrite the integrand:

    2. Integrate term-by-term:

      1. Let .

        Then let and substitute :

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

          1. Let .

            Then let and substitute :

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

              1. The integral of is when :

              So, the result is:

            Now substitute back in:

          So, the result is:

        Now substitute back in:

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

        1. Let .

          Then let and substitute :

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

            1. Let .

              Then let and substitute :

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

                1. The integral of is when :

                So, the result is:

              Now substitute back in:

            So, the result is:

          Now substitute back in:

        So, the result is:

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

        1. Let .

          Then let and substitute :

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

            1. Let .

              Then let and substitute :

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

                1. The integral of is when :

                So, the result is:

              Now substitute back in:

            So, the result is:

          Now substitute back in:

        So, the result is:

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

        1. Let .

          Then let and substitute :

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

            1. Let .

              Then let and substitute :

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

                1. The integral of is when :

                So, the result is:

              Now substitute back in:

            So, the result is:

          Now substitute back in:

        So, the result is:

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

        1. Let .

          Then let and substitute :

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

            1. Let .

              Then let and substitute :

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

                1. The integral of is when :

                So, the result is:

              Now substitute back in:

            So, the result is:

          Now substitute back in:

        So, the result is:

      1. The integral of is .

      The result is:

  2. Now simplify:

  3. Add the constant of integration:


The answer is:

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              
 |                                                                                       
/                                                                                        
$$\int \frac{\left(\log{\left(x \right)} + 1\right)^{5}}{x}\, dx = C + \frac{\log{\left(x \right)}^{6}}{6} + \log{\left(x \right)}^{5} + \frac{5 \log{\left(x \right)}^{4}}{2} + \frac{10 \log{\left(x \right)}^{3}}{3} + \frac{5 \log{\left(x \right)}^{2}}{2} + \log{\left(x \right)}$$
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.