Question

Evaluate:

`int((1 + sinx)/(1 + cosx))e^x dx`

Answer 1

Let I = `int((1 + sinx)/(1 + cosx))e^x dx`

= `inte^x((1 + sinx)/(1 + cosx))dx`

= `inte^x[(1 + 2sin  x/2 cos  x/2)/(2cos^2  x/2)]dx`

= `inte^x[1/(2cos^2  x/2) + (2sin  x/2 cos  x/2)/(2cos^2  x/2)]dx`

= `inte^x[1/2 sec^2  x/2 + tan(x/2)]dx`

Put f(x) = `tan(x/2)`

∴ f'(x) = `d/(dx)[tan  x/2]`

= `sec^2  x/2*1/2`

= `1/2 sec^2  x/2`

∴ I = `inte^x[f(x) + f^'(x)]dx`

= `e^x f(x) + c`

= `e^x * tan(x/2) + c`