Integral of lny dx

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$$\int\limits_{0}^{1} \log{\left(y \right)}\, dy$$

Integral(log(y), (y, 0, 1))

Detail solution

Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

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

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

To find $v{\left(y \right)}$ :
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 1\, dy = y$
Now evaluate the sub-integral.
The integral of a constant is the constant times the variable of integration:

$\int 1\, dy = y$
Now simplify:

$y \left(\log{\left(y \right)} - 1\right)$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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 | log(y) dy = C - y + y*log(y)
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$$\int \log{\left(y \right)}\, dy = C + y \log{\left(y \right)} - y$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

-1

$$-1$$

-1

$$-1$$

-1

Numerical answer [src]

-1.0

-1.0

The graph

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