Question

Evaluate `int_(logsqrt(2))^(logsqrt(3)) 1/((e^x + e^-x)(e^x - e^-x)) dx`.

Answer 1

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 e^2x = t

Then, 2e^2x 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`.