Question

The value of the integral: \[ I = \int_0^3 \frac{dx}{\sqrt{9 - x^2}} \] is:

• A. \( \frac\pi6 \)
• B. \( \frac\pi4 \)
• C. \( \frac\pi2 \)
• D. \( \frac\pi18 \)

Hint:

For integrals of \( \frac1\sqrta^2 - x^2 \), use the arcsine formula.

Answer 1

The Correct Option is C

Step 1: Integral of the given form
The integral \( \int \frac{dx}{\sqrt{a^2 - x^2}} \) is a standard result: \[ \int \frac{dx}{\sqrt{a^2 - x^2}} = \arcsin\left(\frac{x}{a}\right) + C. \]
Step 2: Substitute limits
Here, \( a = 3 \), so: \[ \int_0^3 \frac{dx}{\sqrt{9 - x^2}} = \arcsin\left(\frac{x}{3}\right) \Big|_0^3. \]
Step 3: Evaluate the bounds
\[ \arcsin\left(\frac{3}{3}\right) - \arcsin\left(\frac{0}{3}\right) = \arcsin(1) - \arcsin(0). \] Since \( \arcsin(1) = \frac{\pi}{2} \) and \( \arcsin(0) = 0 \), we get: \[ \frac{\pi}{2} - 0 = \frac{\pi}{2}. \]
Step 4: Verify the options
The correct value is \( \frac{\pi}{2} \), which corresponds to option (C).