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