Mister Exam

Integral of log2x dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

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01log(2x)dx\int\limits_{0}^{1} \log{\left(2 x \right)}\, dx
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    1. Let u=2xu = 2 x.

      Then let du=2dxdu = 2 dx and substitute du2\frac{du}{2}:

      log(u)4du\int \frac{\log{\left(u \right)}}{4}\, du

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

        log(u)2du=log(u)du2\int \frac{\log{\left(u \right)}}{2}\, du = \frac{\int \log{\left(u \right)}\, du}{2}

        1. Use integration by parts:

          udv=uvvdu\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

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

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

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

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

            1du=u\int 1\, du = u

          Now evaluate the sub-integral.

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

          1du=u\int 1\, du = u

        So, the result is: ulog(u)2u2\frac{u \log{\left(u \right)}}{2} - \frac{u}{2}

      Now substitute uu back in:

      xlog(2x)xx \log{\left(2 x \right)} - x

    Method #2

    1. Use integration by parts:

      udv=uvvdu\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

      Let u(x)=log(2x)u{\left(x \right)} = \log{\left(2 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:

        1dx=x\int 1\, dx = x

      Now evaluate the sub-integral.

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

      1dx=x\int 1\, dx = x

  2. Now simplify:

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

  3. Add the constant of integration:

    x(log(2x)1)+constantx \left(\log{\left(2 x \right)} - 1\right)+ \mathrm{constant}


The answer is:

x(log(2x)1)+constantx \left(\log{\left(2 x \right)} - 1\right)+ \mathrm{constant}

The answer (Indefinite) [src]
  /                                
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 | log(2*x) dx = C - x + x*log(2*x)
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2xlog(2x)2x2{{2\,x\,\log \left(2\,x\right)-2\,x}\over{2}}
The answer [src]
-1 + log(2)
2log222{{2\,\log 2-2}\over{2}}
=
=
-1 + log(2)
1+log(2)-1 + \log{\left(2 \right)}
Numerical answer [src]
-0.306852819440055
-0.306852819440055

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