Correct Answer - Option 3 :
\(1 - \frac{1}{{\rm{e}}}\)
Concept:
\(\smallint {{\rm{e}}^{ - {\rm{x}}}}{\rm{dx}} = - {{\rm{e}}^{ - {\rm{x}}}} + {\rm{c}}\)
Calculation:
I = \(\displaystyle\int_0^{\frac{{\rm{\pi }}}{4}} \frac{1}{{{{\rm{e}}^{\tan {\rm{x}}}}\left( {1 - {{\sin }^2}{\rm{x}}} \right)}}{\rm{dx}} \)
\(= {\rm{\;}}\displaystyle\int_0^{\frac{{\rm{\pi }}}{4}} \frac{1}{{{{\rm{e}}^{\tan {\rm{x}}}}{{ \ \cos }^2}{\rm{x}}}}{\rm{dx}} \)
\(= \displaystyle\int_0^{\frac{{\rm{\pi }}}{4}} \frac{{{{\sec }^2}{\rm{x}}}}{{{{\rm{e}}^{\tan {\rm{x}}}}}}{\rm{dx}}\)
Let tan x = t
⇒ sec2 x dx = dt
\({\rm{I}} = \displaystyle\int_0^1 \frac{1}{{{{\rm{e}}^{\rm{t}}}}}{\rm{dt}} \)
\(= \displaystyle\int_0^1 {{\rm{e}}^{ - {\rm{t}}}}{\rm{dt}} \)
\(= \left[ { - {{\rm{e}}^{ - {\rm{t}}}}} \right]_0^1 \)
\(= - \left[ {{{\rm{e}}^{ - 1}} - {{\rm{e}}^0}} \right] \)
\(= 1 - \frac{1}{{\rm{e}}}\)