Question

What is `int_0^(π/2)` sin 2x ℓ n (cot x) dx equal to ?

Options

• 0

• π ℓn 2

• – π ℓn 2

• `(π  ℓn  2)/2`

Answer 1

0

Explanation:

Let I = `int_0^(π/2) sin 2x  ℓn (cot x)dx`

= `int_0^(π/2) sin 2x  ℓn (cos x) dx - int_0^(π/2) sin 2x  ℓn (sin x) dx`

= `int_0^(π/2)  sin[2(π/2 + x)] ℓn cos(π/2 + x)dx - int_0^(π/2) sin  2x  ℓn (sin x) dx`

= `int_0^(π/2) sin(π + 2x) ℓn (sin x)dx - int_0^(π/2) sin  2x  ℓn (sin  x)dx`

= `int_0^(π/2) sin 2x  ℓn (sin  x)dx - int_0^(π/2) sin 2x  ℓn (sin  x)dx`

= 0