Question

What is `int_1^e (lnx)/(2x) dx`?

Answer 1

`= 1/4`

Explanation:

`int_1^e (lnx)/(2x) dx`

`= int_1^e d/dx ( 1/4ln^2x )dx`

`= 1/4[ ln^2x ]_1^e`

`= 1/4[ 1^2 - 0 ]_1^e = 1/4`

Answer 2

`1/4`

Explanation:

Can do this in a number of ways, here are two of them. The first is to use a substitution:

`color(red)("Method 1")`

`int_1^e (ln(x))/(2x) dx = 1/2 int_1^e (ln(x))/(x) dx`

Let `u = ln(x) implies du = (dx)/x`

Transforming the limits:

`u = ln(x) implies u: 0 rarr 1`

Integral becomes:

`1/2int_0^1 u du =1/2 [1/2u^2]_0^1 = 1/2*1/2 = 1/4`

This is the simpler way, but you might not always be able to make a substitution. An alternative is integration by parts.

`color(red)("Method 2")`

Use integration by parts:

For functions `u(x), v(x)`:

`int uv' dx = uv - int u'v dx`

`u(x) = ln(x) implies u'(x) = 1/x`

`v'(x) = 1/(2x) implies v(x) = 1/2ln(x)`

`int (ln(x))/(2x) dx=1/2ln(x)ln(x) - int (ln(x))/(2x) dx`

Grouping like terms:

`2 int (ln(x))/(2x) dx = 1/2ln(x)ln(x) + C`

`therefore int (ln(x))/(2x) dx = 1/4ln(x)ln(x) + C`

We are working with a definite integral though, so applying limits and removing the constant:

`int_(1)^(e) (ln(x))/(2x) dx = [1/4ln(x)ln(x)]_1^e`

`= 1/4ln(e)ln(e) - 1/4ln(1)ln(1)`

`ln(e) = 1, ln(1) = 0`

`implies int_(1)^(e) (ln(x))/(2x) dx = 1/4`