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Question
Evaluate `int_(logsqrt(2))^(logsqrt(3)) 1/((e^x + e^-x)(e^x - e^-x)) dx`.
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Solution
Let I = `int_(logsqrt(2))^(logsqrt(3)) 1/((e^x + e^-x)(e^x - e^-x)) dx`
= `int_(logsqrt(2))^(logsqrt(3)) 1/(((e^(2x) + 1))/e^x xx ((e^(2x) - 1))/e^x) dx`
= `int_(logsqrt(2))^(logsqrt(3)) e^(2x)/((e^(4x) - 1))dx`
Let e2x = t
Then, 2e2x dx = dt
= `int_2^3 dt/(2(t^2 - 1))`
= `1/2 int_2^3 dt/(t^2 - 1^2)`
= `[1/2 xx 1/(2 xx 1) log|(t - 1)/(t + 1)|]_2^3`
= `1/4 [log ((3 - 1)/(3 + 1)) - log ((2 - 1)/(2 + 1))]`
= `1/4 [log 2/4 - log 1/3]`
= `1/4 [log 1/2 + log 3]`
= `1/4 [log 1/2 xx 3]`
= `1/4 log 3/2`.
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