Question

Evaluate the integral \[ \int \frac{dx}{\sqrt{5 + 4x - x^2}} = \]

• A. \( \sin^{-1} \left( \frac{x - 2}{3} \right) + c \)
• B. \( \log \left( |x - 2| + \sqrt{5 + 4x - x^2} \right) + c \)
• C. \( \log \left( |x + 2| + \sqrt{5 + 4x - x^2} \right) + c \)
• D. \( \sin^{-1} \left( \frac{x + 2}{3} \right) + c \)

Hint:

For integrals involving square roots of quadratic expressions, recognize standard forms such as \( \int \frac{dx}{\sqrt{a^2 - x^2}} \), which gives the result involving inverse sine.

Answer 1

The Correct Option is A

Step 1: Understanding the integral.
The given integral has the form \( \int \frac{dx}{\sqrt{a^2 - x^2}} \), which is a standard trigonometric integral. We recognize that we can simplify it using the inverse sine function.

Step 2: Simplifying the expression.
Rearranging the terms inside the square root, we get the expression in the form \( \sin^{-1} \left( \frac{x - 2}{3} \right) \), which corresponds to the standard formula for this type of integral.

Step 3: Conclusion.
Thus, the correct answer is option (A): \[ \sin^{-1} \left( \frac{x - 2}{3} \right) + c \]