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Question
Evaluate the following definite integrals:
`int_0^(pi/2) "e"^x((1 + sin x)/(1 + cos x))"d"x`
Solution
Let I = `int_0^(pi/2) "e"^x((1 + sin x)/(1 + cos x))"d"x`
= `int_0^(pi/2) "e"^x (1/(1 + cosx) + sinx/(1 + cosx))"d"x`
= `int_0^(pi/2) "e"^x (1/(2cos^2 x/2) + (2sin pi/2 cos pi/2)/(2cos^2 pi/2))"d"x`
= `int_0^(pi/2) "e"^x (1/2 sec^2 pi/2 + tan x/2)"d"x`
= `int_0^(pi/2) "e"^x (tan pi/2 + 1/2 sec^2 pi/2)"d"x`
= `["e"^x tan pi/2]_0^(pi/2)`
= `"e"^(pi/2) tan pi/4 - "e"^0 tan 0`
= `"e"^(pi/2) (1) - 0`
I = `"e"^(pi/2)`
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