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: Identify the integral form.
The integral \( \int \frac{dx}{\sqrt{a^2 - x^2}} \) is a known result:
\[ \int \frac{dx}{\sqrt{a^2 - x^2}} = \arcsin\left(\frac{x}{a}\right) + C. \]
Step 2: Apply the limits.
Given \( a = 3 \), the definite integral is:
\[ \int_0^3 \frac{dx}{\sqrt{9 - x^2}} = \arcsin\left(\frac{x}{3}\right) \Big|_0^3. \]
Step 3: Calculate the definite integral.
Evaluate at the upper and lower bounds:
\[ \arcsin\left(\frac{3}{3}\right) - \arcsin\left(\frac{0}{3}\right) = \arcsin(1) - \arcsin(0). \] Using the known values \( \arcsin(1) = \frac{\pi}{2} \) and \( \arcsin(0) = 0 \):
\[ \frac{\pi}{2} - 0 = \frac{\pi}{2}. \]
Step 4: Match with the options.
The computed value is \( \frac{\pi}{2} \), corresponding to option (C).