The value of ∫ 3 6 36 - x 2 3 x 4 d x is equal to

Step-by-step solution

Let, I = ∫ 3 6 36 - x 2 3 x 4 d x Substituting, x = 6 sin ⁡ θ d x = 6 cos ⁡ θ d θ I = ∫ π 6 π 2 36 c o s 2 θ 3 6 4 s i n 4 θ 6 cos ⁡ θ d θ I = ∫ π 6 π 2 c o s 4 θ s i n 4 θ d θ = ∫ π 6 π 2 c o t 2 θ c o s e c 2 θ - 1 d θ = ∫ π 6 π 2 c o t 2 θ c o s e c 2 θ d θ - ∫ π 6 π 2 c o t 2 θ d θ = - [ c o t 3 θ 3 ] π 6 π 2 - ∫ π 6 π 3 ( c o s e c 2 θ - 1 ) d θ = - 1 3 0 - 3 3 + c o t θ π 6 π 2 + θ π 6 π 2 = 3 + 0 - 3 + π 2 - π 6 = π 3