Question

Evaluate \[ \int_{0}^{a} (a-x)^{\frac{3}{2}} x^2 \, dx \]

• A. \(-\dfrac{16a^{9/2}}{315}\)
• B. \(\dfrac{16a^{9/2}}{315}\)
• C. \(\dfrac{16a^{7/2}}{315}\)
• D. \(-\dfrac{16a^{7/2}}{315}\)

Hint:

Definite integrals of the form \(x^m(a-x)^n\) are best solved using Beta functions.

Answer 1

The Correct Option is B

Step 1: Use Beta function property.
The integral is of the form \[ \int_{0}^{a} x^m (a-x)^n dx = a^{m+n+1} \frac{\Gamma(m+1)\Gamma(n+1)}{\Gamma(m+n+2)} \]
Step 2: Identify values.
Here \(m=2\), \(n=\frac{3}{2}\).

Step 3: Substitute values.
\[ \int_{0}^{a} (a-x)^{3/2}x^2 dx = a^{9/2}\frac{\Gamma(3)\Gamma(5/2)}{\Gamma(11/2)} = \frac{16a^{9/2}}{315} \]