Question

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:

• A. \( 40 \)
• B. \( 39 \)
• C. \( 22 \)
• D. \( 26 \)

Hint:

Using substitution simplifies complex integral expressions significantly. Look for substitutions that transform variables into ratios that are easier to integrate.

Answer 1

The Correct Option is B

Step 1: Consider the given integral. \[ I(x) = \int \frac{dx}{(x-11)^{\frac{11}{13}} (x+15)^{\frac{15}{13}}} \] Step 2: Substituting \( t = \frac{x-11}{x+15} \), we get: \[ dt = \frac{26}{(x+5)^2} dx \] Thus, rewriting the integral: \[ I(x) = \frac{1}{26} \int t^{\frac{11}{13}} dt \] Step 3: Solving the integral: \[ I(x) = \frac{1}{26} \times \frac{t^{2/13}}{2/13} \] Step 4: Evaluating \( I(x) \): \[ I(x) = \frac{1}{4} \left( \frac{x-11}{x+15} \right)^{2/13} + C \] Step 5: Computing \( I(37) - I(24) \): \[ I(37) - I(24) = \frac{1}{4} \left( \left( \frac{26}{52} \right)^{2/13} - \left( \frac{13}{39} \right)^{2/13} \right) \] Step 6: Expressing in terms of \( b \) and \( c \): \[ = \frac{1}{4} \left( \frac{1}{2^{2/13}} - \frac{1}{3^{2/13}} \right) \] \[ = \frac{1}{4} \left( \frac{1}{4^{1/13}} - \frac{1}{9^{1/13}} \right) \] Thus, \( b = 4 \) and \( c = 9 \). Final Step: Calculating \( 3(b+c) \): \[ 3(4+9) = 39 \]