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Question
Evaluate: `int "e"^"x"/(4"e"^"2x" -1)` dx
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Solution
Let I = `int "e"^"x"/(4"e"^"2x" -1)` dx
`"I" = int "e"^"x"/(4("e"^"x")^2 - 1)` dx
Put ex = t
∴ ex dx = dt
∴ I = `int "dt"/(4"t"^2 - 1)`
`∴ "I" = 1/4 int 1/("t"^2 - 1/4)` dt
`∴ "I" = 1/4 int 1/("t"^2 - (1/2)^2)` dt
`∴ "I" = 1/4 . 1/(2 (1/2)) log |("t" - 1/2)/("t" + 1/2)|` + c
`∴ "I" = 1/4 log |("2t" - 1)/("2t" + 1)|` + c
Resubstitute t = ex
`∴ "I" = 1/4 log |(2"e"^"x" - 1)/(2"e"^"x" + 1)|` + c
Notes
Answer in the textbook is incorrect.
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