Question

Evaluate : `int_(-1)^1 (1)/(a^2e^x + b^2e^(-x))*dx`

Answer 1

Let I = `int_(-1)^1 e^x/(a^2(e^x)^2 + b^2)*dx`
Put e^x = t
∴ e^x·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")]`.