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Solving integrals/ ln^2(x-3)/(x-3)

Integral of ln^2(x-3)/(x-3) dx

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The solution

You have entered [src] 
  1               
  /               
 |                
 |     2          
 |  log (x - 3)   
 |  ----------- dx
 |     x - 3      
 |                
/                 
0
∫01log⁡(x−3)2x−3 dx\int\limits_{0}^{1} \frac{\log{\left(x - 3 \right)}^{2}}{x - 3}\, dx 
Integral(log(x - 3)^2/(x - 3), (x, 0, 1))
Detail solution

  1. There are multiple ways to do this integral.

    Method #1

    1. Let u=log⁡(x−3)u = \log{\left(x - 3 \right)}.

      Then let du=dxx−3du = \frac{dx}{x - 3} and substitute dudu:

      ∫u2 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}

      Now substitute uu back in:

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

    Method #2

    1. Let u=x−3u = x - 3.

      Then let du=dxdu = dx and substitute dudu:

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

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

        Then let du=−duu2du = - \frac{du}{u^{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⁡(u)33\frac{\log{\left(u \right)}^{3}}{3}

      Now substitute uu back in:

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

  2. Now simplify:

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

  3. Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src] 
  /                                
 |                                 
 |    2                    3       
 | log (x - 3)          log (x - 3)
 | ----------- dx = C + -----------
 |    x - 3                  3     
 |                                 
/
∫log⁡(x−3)2x−3 dx=C+log⁡(x−3)33\int \frac{\log{\left(x - 3 \right)}^{2}}{x - 3}\, dx = C + \frac{\log{\left(x - 3 \right)}^{3}}{3}
Simplify
The graph
-0.010-0.008-0.006-0.004-0.0020.0100.0000.0020.0040.0060.0080.00
Plot the graph f(x)
The answer [src] 
                 3                  3
  (pi*I + log(3))    (pi*I + log(2)) 
- ---------------- + ----------------
         3                  3
(log⁡(2)+iπ)33−(log⁡(3)+iπ)33\frac{\left(\log{\left(2 \right)} + i \pi\right)^{3}}{3} - \frac{\left(\log{\left(3 \right)} + i \pi\right)^{3}}{3} 
=
= 
                 3                  3
  (pi*I + log(3))    (pi*I + log(2)) 
- ---------------- + ----------------
         3                  3
(log⁡(2)+iπ)33−(log⁡(3)+iπ)33\frac{\left(\log{\left(2 \right)} + i \pi\right)^{3}}{3} - \frac{\left(\log{\left(3 \right)} + i \pi\right)^{3}}{3} 
-(pi*i + log(3))^3/3 + (pi*i + log(2))^3/3
Numerical answer [src] 
(3.67079877942085 - 2.28235432962615j)
(3.67079877942085 - 2.28235432962615j)

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

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