∫ 0 5 cos π x - x 2 d x , where t denotes greatest integer less than or equal to t , is equal to

Step-by-step solution

I = ∫ 0 5 cos x 2 π - π x d x ∵ x ∈ 0 , 5   ∴ x ∈ 0 , 5 2 ⇒ x 2 ∈ 0 , 1 , 2 I = ∫ 0 2 cos - π x d x + ∫ 2 4 cos π - π x d x + ∫ 4 5 cos 2 π - π x d x ⇒ I = sinπ x π 0 2 + - sinπ x π 2 4 + sinπ x π 4 5 = 0