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Calculate integral int_1^e 1/x((ln x)^2+1)dx?

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ruals | (Level 1) Salutatorian

Posted September 25, 2013 at 10:26 AM via web

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Calculate integral int_1^e 1/x((ln x)^2+1)dx?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted September 25, 2013 at 10:58 AM (Answer #1)

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You need to evaluate the given definite integral, hence, you should use the integration by substitution, such that:

`ln x = t => (1/x)dx = dt`

You need to change the limits of integration, such that:

`x = 1 => ln 1 = 0 => t = 0`

`x = e => ln e = 1 => t = 1`

Changing the variable and the limits of integration yields:

`int_0^1 (dt)/(t^2+1) = tan^(-1) t|_0^1`

Using the fudamental theorem of calculus, yields:

`int_0^1 (dt)/(t^2+1) = tan^(-1) 1 - tan^(-1) 0`

`int_0^1 (dt)/(t^2+1) = pi/4 - 0`

`int_0^1 (dt)/(t^2+1) = pi/4`

Hence, evaluating the given definite integral, using integration by substitution, yields `int_1^e 1/x((ln x)^2+1) dx = pi/4` .

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