Integral of 1/(2y) dY

The solution

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$$\int\limits_{0}^{1} \frac{1}{2 y}\, dy$$

Integral(1/(2*y), (y, 0, 1))

Detail solution

Let $u = 2 y$ .

Then let $du = 2 dy$ and substitute $\frac{du}{2}$ :

$\int \frac{1}{2 u}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{1}{u}\, du = \frac{\int \frac{1}{u}\, du}{2}$
  1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
  So, the result is: $\frac{\log{\left(u \right)}}{2}$
Now substitute $u$ back in:

$\frac{\log{\left(2 y \right)}}{2}$
Add the constant of integration:

$\frac{\log{\left(2 y \right)}}{2}+ \mathrm{constant}$

The answer is:

$\frac{\log{\left(2 y \right)}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                     
 |                      
 |  1           log(2*y)
 | --- dy = C + --------
 | 2*y             2    
 |                      
/

$$\int \frac{1}{2 y}\, dy = C + \frac{\log{\left(2 y \right)}}{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

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$$\infty$$

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$$\infty$$

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Numerical answer [src]

22.0452230669964

22.0452230669964

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

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Integral of 1/(2y) dY

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The solution