lim x → π 2 1 x - π 2 2 ∫ x 3 π 2 3 cos 1 t 3 d t is equal to

Step-by-step solution

Let, y = lim x → π 2 ∫ x 3 π 2 3 cos t 1 3 d t x - π 2 2 ⇒ y = lim h → 0 ∫ π 2 - h 3 π 2 3 cos t 1 3 d t π 2 - h - π 2 2 ⇒ y = lim h → 0 ∫ π 2 - h 3 π 2 3 cos t 1 3 d t h 2 Applying L-Hospital's rule and Newton Leibnitz Theorem we get, ⇒ y = lim h → 0 0 + cos π 2 - h × 3 × π 2 - h 2 2 h ⇒ y = lim h → 0 sin h × 3 × π 2 - h 2 2 h We know that, lim h → 0 sin h h = 1 ⇒ y = lim h → 0 3 × π 2 - h 2 2 ⇒ y = 3 π 2 8