Mister Exam

Other calculators

Solving integrals/ log2(x)

Integral of log2(x) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

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

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

    ∫log⁡(x)log⁡(2) dx=∫log⁡(x) dxlog⁡(2)\int \frac{\log{\left(x \right)}}{\log{\left(2 \right)}}\, dx = \frac{\int \log{\left(x \right)}\, dx}{\log{\left(2 \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⁡(2)\frac{x \log{\left(x \right)} - x}{\log{\left(2 \right)}}

  2. Now simplify:

    x(log⁡(x)−1)log⁡(2)\frac{x \left(\log{\left(x \right)} - 1\right)}{\log{\left(2 \right)}}

  3. Add the constant of integration:

    x(log⁡(x)−1)log⁡(2)+constant\frac{x \left(\log{\left(x \right)} - 1\right)}{\log{\left(2 \right)}}+ \mathrm{constant}

The answer is:

x(log⁡(x)−1)log⁡(2)+constant\frac{x \left(\log{\left(x \right)} - 1\right)}{\log{\left(2 \right)}}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                             
 |                              
 | log(x)          -x + x*log(x)
 | ------ dx = C + -------------
 | log(2)              log(2)   
 |                              
/
x log⁡x−xlog⁡2{{x\,\log x-x}\over{\log 2}}
Simplify
The answer [src] 
 -1   
------
log(2)
−1log⁡2-{{1}\over{\log 2}} 
=
= 
 -1   
------
log(2)
−1log⁡(2)- \frac{1}{\log{\left(2 \right)}}
Simplify
Numerical answer [src] 
-1.44269504088896
-1.44269504088896

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

    © Mister Exam – Calculator