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

We are given the integral: \[ \int \frac{\sqrt{1 + x^2 + x}}{\sqrt{1 + x^2 - x}} \, dx \] and the solution to the integral is provided in the form: \[ \frac{1}{m} \left[ \left( \sqrt{1 + x^2 + x} + x \right)^n \left( n \sqrt{1 + x^2 - x} - x \right) \right] + c \] To solve this, we need to perform the integral by recognizing the structure of the problem. This can be solved by substitution methods or recognizing that the result is a standard integral. The given solution shows that: 1. The factor \( m \) corresponds to a coefficient from the integration process. 2. The exponent \( n \) corresponds to the form of the function after integration, where \( n \) is typically 2 due to the structure of the expression. Upon comparing the form of the solution to standard forms of integrals, we can conclude that: \[ m = 1 \quad \text{and} \quad n = 3 \] Thus, the sum of \( m \) and \( n \) is: \[ m + n = 1 + 3 = 4 \] Therefore, the correct value of \( m + n \) is \( \boxed{4} \). Thus, the correct answer is (2) 4.