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Integral of x(1-ln(x))²+1 dx

Limits of integration:

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The graph:

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The solution

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  3                         
  /                         
 |                          
 |  /              2    \   
 |  \x*(1 - log(x))  + 1/ dx
 |                          
/                           
1                           
$$\int\limits_{1}^{3} \left(x \left(1 - \log{\left(x \right)}\right)^{2} + 1\right)\, dx$$
Integral(x*(1 - log(x))^2 + 1, (x, 1, 3))
Detail solution
  1. Integrate term-by-term:

    1. There are multiple ways to do this integral.

      Method #1

      1. Rewrite the integrand:

      2. Integrate term-by-term:

        1. Let .

          Then let and substitute :

          1. Use integration by parts:

            Let and let .

            Then .

            To find :

            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 the exponential function is itself.

                So, the result is:

              Now substitute back in:

            Now evaluate the sub-integral.

          2. Use integration by parts:

            Let and let .

            Then .

            To find :

            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 the exponential function is itself.

                So, the result is:

              Now substitute back in:

            Now evaluate the sub-integral.

          3. 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 the exponential function is itself.

                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. Use integration by parts:

              Let and let .

              Then .

              To find :

              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 the exponential function is itself.

                  So, the result is:

                Now substitute back in:

              Now evaluate the sub-integral.

            2. 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 the exponential function is itself.

                  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 when :

        The result is:

      Method #2

      1. Rewrite the integrand:

      2. Integrate term-by-term:

        1. Let .

          Then let and substitute :

          1. Use integration by parts:

            Let and let .

            Then .

            To find :

            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 the exponential function is itself.

                So, the result is:

              Now substitute back in:

            Now evaluate the sub-integral.

          2. Use integration by parts:

            Let and let .

            Then .

            To find :

            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 the exponential function is itself.

                So, the result is:

              Now substitute back in:

            Now evaluate the sub-integral.

          3. 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 the exponential function is itself.

                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. Use integration by parts:

              Let and let .

              Then .

              To find :

              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 the exponential function is itself.

                  So, the result is:

                Now substitute back in:

              Now evaluate the sub-integral.

            2. 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 the exponential function is itself.

                  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 when :

        The result is:

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

    The result is:

  2. Now simplify:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                                  
 |                                       2    2    2         2       
 | /              2    \              5*x    x *log (x)   3*x *log(x)
 | \x*(1 - log(x))  + 1/ dx = C + x + ---- + ---------- - -----------
 |                                     4         2             2     
/                                                                    
$$\int \left(x \left(1 - \log{\left(x \right)}\right)^{2} + 1\right)\, dx = C + \frac{x^{2} \log{\left(x \right)}^{2}}{2} - \frac{3 x^{2} \log{\left(x \right)}}{2} + \frac{5 x^{2}}{4} + x$$
The graph
The answer [src]
                      2   
     27*log(3)   9*log (3)
12 - --------- + ---------
         2           2    
$$- \frac{27 \log{\left(3 \right)}}{2} + \frac{9 \log{\left(3 \right)}^{2}}{2} + 12$$
=
=
                      2   
     27*log(3)   9*log (3)
12 - --------- + ---------
         2           2    
$$- \frac{27 \log{\left(3 \right)}}{2} + \frac{9 \log{\left(3 \right)}^{2}}{2} + 12$$
12 - 27*log(3)/2 + 9*log(3)^2/2
Numerical answer [src]
2.60000442663714
2.60000442663714

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