Question

The value of `int_0^(pi/2) log ((4 + 3 sin x)/(4 + 3 cos x))  dx` is

Options

• 2

• `3/4`

• 0

• – 2

Answer 1

0

Explanation:

Let I = `int_0^(pi/2) log ((4 + 3 sin x)/(4 + 3 cos x))  dx`  ......(1)

= `int_0^(pi/2) log ((4 + 3 sin (pi/2 - x))/(4 + 3 cos (pi/2 - x))) dx`

= `[because  int_0^a f(x) dx = int_0^a f(a - x)  dx]`

I = `int_0^(pi/2) log  (4 + 3 cos x)/(4 + 3 sin x)  dx`  ......(2)

Adding (1) and (2)

2I = `int_0^(pi/2) (log  (4 + 3 sin x)/(4 + 3 cos x)  dx + log  (4 + 3 cos x)/(4 + 3 sin x)  dx)`

= `int_0^(pi/2) log  ((4 + 3 sin x)/(4 + 3 cos x) xx (4 + 3 cos x)/(4 + 3 sin x)) dx`

= `int_0^(pi/2) log 1  dx`

= 0