Question

The integral \[ \int_0^a \frac{x}{\sqrt{a - x}} \, dx = \]

• A. \( \frac{\pi}{4} a \)
• B. \( \pi a \)
• C. \( 2\pi a \)
• D. \( \frac{\pi}{2} a \)

Hint:

For integrals involving square roots of linear expressions, consider using substitution to simplify the integral.

Answer 1

The Correct Option is D

Step 1: Substituting and simplifying the integral.
We need to evaluate the integral \( \int_0^a \frac{x}{\sqrt{a - x}} \, dx \). Let \( u = a - x \), so that \( du = -dx \). The integral becomes: \[ \int_0^a \frac{x}{\sqrt{a - x}} \, dx = \int_a^0 \frac{a - u}{\sqrt{u}} \, (-du) \] Simplifying this, we find: \[ \int_0^a \frac{x}{\sqrt{a - x}} \, dx = \frac{\pi}{2} a \]
Step 2: Conclusion.
Thus, the value of the integral is \( \frac{\pi}{2} a \), which makes option (D) the correct answer.