Question

Evaluate the following definite integrals:

`int_0^(pi/2) "e"^x((1 + sin x)/(1 + cos x))"d"x`

Answer 1

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)`