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Solving integrals/ loge(x)

Integral of loge(x) dx

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src] 
  E           
  /           
 |            
 |   log(x)   
 |  ------- dx
 |     / 1\   
 |  log\e /   
 |            
/             
1
∫1elog⁡(x)log⁡(e1) dx\int\limits_{1}^{e} \frac{\log{\left(x \right)}}{\log{\left(e^{1} \right)}}\, dx 
Integral(log(x)/log(exp(1)), (x, 1, E))
Detail solution

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

    ∫log⁡(x)log⁡(e1) dx=∫log⁡(x) dxlog⁡(e1)\int \frac{\log{\left(x \right)}}{\log{\left(e^{1} \right)}}\, dx = \frac{\int \log{\left(x \right)}\, dx}{\log{\left(e^{1} \right)}}

    1. Use integration by parts:

      ∫u⁡dv⁡=u⁡v⁡−∫v⁡du⁡\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

      Let u(x)=log⁡(x)u{\left(x \right)} = \log{\left(x \right)} and let dv⁡(x)=1\operatorname{dv}{\left(x \right)} = 1.

      Then du⁡(x)=1x\operatorname{du}{\left(x \right)} = \frac{1}{x}.

      To find v(x)v{\left(x \right)}:

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

        ∫1 dx=x\int 1\, dx = x

      Now evaluate the sub-integral.

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

      ∫1 dx=x\int 1\, dx = x

    So, the result is: xlog⁡(x)−xlog⁡(e1)\frac{x \log{\left(x \right)} - x}{\log{\left(e^{1} \right)}}

  2. Now simplify:

    x(log⁡(x)−1)x \left(\log{\left(x \right)} - 1\right)

  3. Add the constant of integration:

    x(log⁡(x)−1)+constantx \left(\log{\left(x \right)} - 1\right)+ \mathrm{constant}

The answer is:

x(log⁡(x)−1)+constantx \left(\log{\left(x \right)} - 1\right)+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                              
 |                               
 |  log(x)          -x + x*log(x)
 | ------- dx = C + -------------
 |    / 1\                / 1\   
 | log\e /             log\e /   
 |                               
/
∫log⁡(x)log⁡(e1) dx=C+xlog⁡(x)−xlog⁡(e1)\int \frac{\log{\left(x \right)}}{\log{\left(e^{1} \right)}}\, dx = C + \frac{x \log{\left(x \right)} - x}{\log{\left(e^{1} \right)}}
Simplify
The graph
1.01.21.41.61.82.02.22.42.62-2
Plot the graph f(x)
The answer [src] 
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11 
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Numerical answer [src] 
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    Use the examples entering the upper and lower limits of integration.

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