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Calculate int_(1/e)^e |lnx|/xdx?

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ruals | Salutatorian

Posted September 24, 2013 at 10:14 AM via web

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Calculate int_(1/e)^e |lnx|/xdx?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted September 24, 2013 at 10:27 AM (Answer #1)

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You need to use absolute value definition, such that:

`|ln x| = {(ln x, x in [1;e]),(-ln x, x in [1/e;1]):}`

You need to use the following additivity convention of definite integrals, such that:

`int_a^b f(x) dx = int_a^c f(x)dx + int_c^b f(x)dx`

Reasoning by analogy, yields:

`int_(1/e)^e (|ln x|)/x dx = int_(1/e)^1 (-ln x)/x dx + int_1^e (ln x)/x dx`

You should solve the integrals using integration by substitution, such that:

`ln x = y => 1/x dx = dy`

`x = 1/e => ln (1/e) = -1 = y`

`x = 1 => ln 1 = y => y = 0`

`int_(1/e)^e (|ln x|)/x dx = -int_(-1)^0 y dy + int_0^1 ydy`

`int_(1/e)^e (|ln x|)/x dx = - y^2/2|_(-1)^0 + y^2/2|_0^1`

Using the fundamental theorem of calculus, yields:

`int_(1/e)^e (|ln x|)/x dx = - (0^2/2 - (-1)^2/2) + (1^2/2 - 0^2/2)`

`int_(1/e)^e (|ln x|)/x dx = 1/2 + 1/2 = 1`

Hence, evaluating the given definite integral, using absolute value definition and the properties of definite integral, yields `int_(1/e)^e (|ln x|)/x dx = 1.`

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