Question

Evaluate:

`int_0^(pi//2)(5 sin x + 3 cos x)/(sin x + cos x)  dx`

Answer 1

`int_0^(pi//2)(5 sin x + 3 cos x)/(sin x + cos x)  dx    ...(i)`

By using property of definite integral

`int_a^b f(x)dx`

= `int_a^b f(a + b - x)dx`

We get

I = `int_0^(pi//2)(5 sin(pi/2 - x) + 3 cos (pi/2 - x))/(sin(pi/2 - x) + cos(pi/2 - x))  dx`

`I = int_0^(pi//2)(5 cos x + 3 sin x)/(cos x + sin x)  dx    ...(ii)`

On Adding (i) & (ii) 

= `int_0^(pi//2)(5 sin x + 3 cos x)/(sin x + cos x) + (5 cos x + 3 sin x)/(cos x + sin x)` 

= `int_0^(pi//2)(5(sin x + cos x) + 3(sin x + cos x))/(sin x + cos x)`

= `int_0^(pi//2) (8(sin x + cos x))/(sin x + cos x)  dx`

= `int_0^(pi//2) 8  dx`

= `8(x)_0^(pi//2)`

= `(8pi)/2`

∴ 2I = 4π

I = `(4pi)/2`

∴ I = 2π