If f x = ∫ 0 x t sin x - sin t d t , then

Step-by-step solution

Given f x = ∫ 0 x t sin x - sin t d t ⇒ f x = sin x ∫ 0 x t d t - ∫ 0 x t sin t d t On differentiating, we get ⇒ f ' x = sin x x + cos x ∫ 0 x t d t - x sin x ⇒ f ' x = cos x ∫ 0 x t d t Differentiating the above equation again, we get ⇒ f '' x = cos x x - sin x ∫ 0 x t d t ⇒ f ''' x = x - sin x + cos x - sin x x - cos x ∫ 0 x t d t ⇒ f ''' x + f ' x = cos x - 2 x sin x .