Question

\[ \int_9^{10} \frac{\sqrt{1 + x^2 + x}}{\sqrt{1 + x^2 - x}} \, dx = \frac{1}{m} \left[ \left( \sqrt{1 + x^2 + x} + x \right)^n \left( n \sqrt{1 + x^2 - x} - x \right) \right] + c \] where \( c \) is the constant of integration and \( m, n \in \mathbb{N} \), then \( m + n \) is ______.

• A. 3
• B. 4
• C. 5
• D. 6

Hint:

The method of substitution and recognizing standard integrals help in identifying the constants of integration.

Answer 1

The Correct Option is B

The integral provided is: \[ \int \frac{\sqrt{1 + x^2 + x}}{\sqrt{1 + x^2 - x}} \, dx \] The solution format is: \[ \frac{1}{m} \left[ \left( \sqrt{1 + x^2 + x} + x \right)^n \left( n \sqrt{1 + x^2 - x} - x \right) \right] + c \] Solving this integral requires recognizing its structure, potentially using substitution or identifying it as a standard integral form. The provided solution indicates: 1. \( m \) is a coefficient arising from the integration. 2. \( n \) is an exponent related to the integrated function's form, typically 2 based on the expression's structure. Comparing the solution's form with known integral forms leads to the conclusion: \[ m = 1 \quad \text{and} \quad n = 3 \] The sum of \( m \) and \( n \) is calculated as: \[ m + n = 1 + 3 = 4 \] The final value of \( m + n \) is \( \boxed{4} \). The correct answer is (2) 4.