Question

Evaluate :`int_0^(pi/2)dx/(1+cotx)`

Answer 1

Let `I=int_0^(pi/2)dx/(1+cotx)`

` =int_0^(pi/2)dx/(1+(cosx)/(sinx))`

`=int_0^(pi/2)dx/((sinx+cosx)/sinx)`

`=int_0^(pi/2)sinx/(sinx+cosx)dx..................(i)`

We know 

`int_0^af(x)dx=int_0^af(a-x)dx`

`I=int_0^(pi/2)(sin(pi/2-x))/(sin(pi/2-x)+cos(pi/2-x))`

`I=int_0^(pi/2)cosx/(cosx+sinx)dx.................(ii)`

Adding (i) and (ii)

`I+I=int_0^(pi/2)sinx/(sinx+cosx)dx+int_0^(pi/2)cosx/(cosx+sinx)dx`

`2I=int_0^(pi/2)(sinx+cosx)/(sinx+cosx)dx`

`2I=int_0^(pi/2)1dx=[x]_0^(pi/2)`

`I=1/2[x]_0^(pi/2)=1/2[pi/2-0]`

`I=pi/4`