Mister Exam

Integral of ln(y) dx

Limits of integration:

from to
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The graph:

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Piecewise:

The solution

You have entered [src]
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01log(y)dy\int\limits_{0}^{1} \log{\left(y \right)}\, dy
Integral(log(y), (y, 0, 1))
Detail solution
  1. Use integration by parts:

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

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

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

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

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

      1dy=y\int 1\, dy = y

    Now evaluate the sub-integral.

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

    1dy=y\int 1\, dy = y

  3. Now simplify:

    y(log(y)1)y \left(\log{\left(y \right)} - 1\right)

  4. Add the constant of integration:

    y(log(y)1)+constanty \left(\log{\left(y \right)} - 1\right)+ \mathrm{constant}


The answer is:

y(log(y)1)+constanty \left(\log{\left(y \right)} - 1\right)+ \mathrm{constant}

The answer (Indefinite) [src]
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log(y)dy=C+ylog(y)y\int \log{\left(y \right)}\, dy = C + y \log{\left(y \right)} - y
The graph
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The answer [src]
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Numerical answer [src]
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The graph
Integral of ln(y) dx

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