Step 1: Use the substitution Let \[ t = \sin^{-1} \left( \sqrt{\frac{x - a}{x}} \right). \] Then, \[ \sin t = \sqrt{\frac{x - a}{x}}. \] Squaring both sides, \[ \sin^2 t = \frac{x - a}{x}. \] Thus, \[ x \sin^2 t = x - a. \] Step 2: Differentiate both sides Using implicit differentiation, \[ 2x \sin t \cos t \frac{dt}{dx} + \sin^2 t = 1. \] Rearranging, \[ \frac{dt}{dx} = \frac{1 - \sin^2 t}{2x \sin t \cos t}. \] Using trigonometric identities, solve the integral, leading to: \[ x \cos^{-1} \left(\sqrt{\frac{a}{x}}\right) - \sqrt{ax - a^2} + C. \]
Let \[ I(x) = \int \frac{dx}{(x-11)^{\frac{11}{13}} (x+15)^{\frac{15}{13}}} \] If \[ I(37) - I(24) = \frac{1}{4} \left( b^{\frac{1}{13}} - c^{\frac{1}{13}} \right) \] where \( b, c \in \mathbb{N} \), then \[ 3(b + c) \] is equal to: