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What is integral (ln 2 to ln 3) 1/(e^x - 1)dx ?

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yapayapa | (Level 2) Honors

Posted July 8, 2013 at 4:31 PM via web

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What is integral (ln 2 to ln 3) 1/(e^x - 1)dx ?

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llltkl | College Teacher | (Level 3) Valedictorian

Posted July 8, 2013 at 5:03 PM (Answer #1)

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Given definite integral is `int_ln2^ln3(1/(e^x - 1))dx`

`=int_ln2^ln3(e^x-(e^x-1))/(e^x - 1))dx`

`=int_ln2^ln3(e^x)/(e^x-1)dx -int_ln2^ln3(e^x-1)/(e^x-1)dx`

`=int_ln2^ln3(d(e^x-1))/(e^x-1)dx -int_ln2^ln3 1dx`

`[ln(e^x-1)]_(ln2)^(ln3)-[x]_ln2^ln3`

=`[ln(e^(ln3)-1)-ln(e^(ln2)-1)]-[ln3-ln2]`

`=ln(3-1)-ln(2-1)-ln3+ln2`

`=2ln2-ln3`

`=ln4-ln3`

`=ln(4/3)`

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