Question

Evaluate: \(\displaystyle \int \sqrt{a^2 - x^2} \, dx = \)

• A. \( 2x \sqrt{a^2 - x^2} + c \)
• B. \( \frac{x}{2} \sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{-1} \frac{x}{a} + c \)
• C. \( 2x \sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{-1} \frac{x}{a} + c \)
• D. \( 2x \sqrt{x^2 - a^2} - \frac{a^2}{2} \sin^{-1} \frac{x}{a} + c \)

Hint:

For integrals involving \(\sqrt{a^2 - x^2}\), use the formula: \[ \int \sqrt{a^2 - x^2} \, dx = \frac{x}{2} \sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{-1} \frac{x}{a} + c \] Alternatively, try the substitution \( x = a \sin \theta \).

Answer 1

The Correct Option is B

Step 1: Recall the formula: \[ \int \sqrt{a^2 - x^2} \, dx = \frac{x}{2} \sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{-1} \frac{x}{a} + c \] Step 2: This can be derived by using integration by parts or trigonometric substitution.