Question

If ∫f(sin x)dx for x ∈ [0, π] = 2 then the value of ∫xf(sin x)dx for x ∈ [0, π] is........ 

(a) 0 

(b) 4 

(c) (π/2) 

(d) π

Answer 1

The correct option (d) π  

Explanation:

^π∫₀f(sinx)dx = 2 

I = ^π∫₀x ∙ f(sin x)dx 

= ^π∫₀(π – x)f[sin(π – x)]dx 

= ^π∫₀(π – x)f(sin x)dx 

I = π ^π∫₀f(sinx)dx – ^π∫₀ x ∙ f(sin x)dx 

∴ 2I = π ^π∫₀f(sin x)dx 

∴ 2I = 2π 

I = π.