If ∫ d x x 2 - 2 x + 10 2 = A tan - 1 x - 1 3 + f x x 2 - 2 x + 10 + C , then (where C is a constant of integration)

Step-by-step solution

We have I = ∫ d x x 2 - 2 x + 10 2 = ∫ d x x - 1 2 + 3 2 2 Let x   – 1 = 3 tan θ ⇒ d x = 3 sec 2 θ d θ ⇒ I = ∫ 3 s e c 2 θ d θ 81 s e c 4 θ = 1 27 ∫ c o s 2 θ d θ = 1 54 ∫ 1 + cos 2 θ d θ = 1 54 θ + sin 2 θ 2 + C = 1 54 θ + sin θ cos θ + C , where C is the constant of integration. = 1 54 tan - 1 x - 1 3 + x - 1 3 1 + x - 1 3 2 + C = 1 54 t a n - 1 x - 1 3 + 3 x - 1 x 2 - 2 x + 10 + C Hence