Question

Let \[ P = \int_0^1 \frac{dx}{\sqrt{8 - x^2 - x^3}}. \] Which of the following statements is TRUE?

• A. \( \sin^{-1} \left( \frac{1}{\sqrt{2}} \right)<P<\sin^{-1} \left( \frac{1}{\sqrt{2}} \right) \)
• B. \( \frac{1}{\sqrt{2}} \sin^{-1} \left( \frac{1}{\sqrt{2}} \right)<P<\sin^{-1} \left( \frac{1}{2} \right) \)
• C. \( \frac{1}{\sqrt{2}} \sin^{-1} \left( \frac{1}{\sqrt{2}} \right)<P<\sin^{-1} \left( \frac{1}{2} \right) \)
• D. \( \sin^{-1} \left( \frac{1}{\sqrt{2}} \right)<P<\frac{\sqrt{3}}{2} \sin^{-1} \left( \frac{1}{2} \right) \)

Hint:

When evaluating integrals with trigonometric terms, look for standard forms involving inverse trigonometric functions or simplifications to reduce the integral.

Answer 1

The Correct Option is A

Step 1: Analyze the integral expression.
The integral involves a square root term in the denominator and resembles the structure of an inverse trigonometric function. To compute this integral, we use known integral forms for trigonometric identities.
Step 2: Final value of the integral.
The integral evaluates to a value \( P \) between the limits \( \sin^{-1} \left( \frac{1}{\sqrt{2}} \right) \) and \( \sin^{-1} \left( \frac{1}{\sqrt{2}} \right) \), which is consistent with option (A).