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Question
Evaluate: `int_(-1)^1 1/("a"^2"e"^x + "b"^2"e"^(-x)) "d"x`
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Solution
Let I = `int_(-1)^1 1/("a"^2"e"^x + "b"^2"e"^(-x)) "d"x`
= `int_(-1)^1 1/("a"^2"e"^x + ("b"^2)/("e"^x)) "d"x`
= `int_(-1)^1 "e"^x/("a"^2("e"^x)^2 + "b"^2) "d"x`
Put ex = t
∴ ex dx = dt
When x = −1, t = e−1 and when x = 1, t = e
∴ I = `int_("e"^-1)^"e" "dt"/("a"^2"t"^2 + "b"^2)`
= `1/("a"^2) int_("e"^-1)^"e" "dt"/("t"^2 + ("b"/"a")^2`
= `1/("a"^2)[1/("b"/"a")tan^-1 ("t"/("b"/"a"))]_("e"^-1)^"e"`
= `1/"ab"[tan^-1("at"/"b")]_("e"^-1)^"e"`
∴ I = `1/"ab"[tan^-1("ae"/"b") - tan^-1("a"/"be")]`
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