Question

The value of the integral \[ \int_{-2}^{2} x^4 (4 - x^2)^{\frac{7}{2}} \, dx \] is equal to:

• A. \( 4\pi \)
• B. \( \frac{\pi}{16} \)
• C. \( 28\pi \)
• D. \( \frac{3\pi}{128} \)

Hint:

For integrals involving \( (a^2 - x^2)^{n/2} \) over symmetric limits, always check for the Even/Odd property first. If even, the transformation \( x = a\sin\theta \) will almost always lead to a Wallis' Formula application.

Answer 1

The Correct Option is C

Concept: This integral involves a symmetric interval and a radical form \( \sqrt{a^2 - x^2} \). The most effective approach is:
• Even Function Property: If \( f(-x) = f(x) \), then \( \int_{-a}^{a} f(x) \, dx = 2 \int_{0}^{a} f(x) \, dx \).
• Trigonometric Substitution: Replacing \( x \) with \( a \sin \theta \) to simplify the radical expression.
• Wallis' Formula: To evaluate the resulting integral of the form \( \int_{0}^{\pi/2} \sin^m \theta \cos^n \theta \, d\theta \).

Step 1: Checking symmetry and simplifying limits.
The integrand \( f(x) = x^4 (4 - x^2)^{7/2} \) is an even function because replacing \( x \) with \( -x \) leaves the expression unchanged (due to the even powers 4 and 2). \[ I = 2 \int_{0}^{2} x^4 (4 - x^2)^{7/2} \, dx \]

Step 2: Applying trigonometric substitution.
Let \( x = 2\sin \theta \), then \( dx = 2\cos \theta \, d\theta \).
• When \( x = 0, \theta = 0 \).
• When \( x = 2, \theta = \pi/2 \). The term \( (4 - x^2)^{7/2} = (4 - 4\sin^2 \theta)^{7/2} = (4\cos^2 \theta)^{7/2} = (2^2)^{7/2} (\cos^2 \theta)^{7/2} = 2^7 \cos^7 \theta \). Substituting into the integral: \[ I = 2 \int_{0}^{\pi/2} (2\sin \theta)^4 (2^7 \cos^7 \theta) (2\cos \theta) \, d\theta \] \[ I = 2 \times 2^4 \times 2^7 \times 2 \int_{0}^{\pi/2} \sin^4 \theta \cos^8 \theta \, d\theta = 2^{13} \int_{0}^{\pi/2} \sin^4 \theta \cos^8 \theta \, d\theta \]

Step 3: Evaluating using Wallis' Formula.
For \( m=4, n=8 \), both are even, so the formula is: \[ \int_{0}^{\pi/2} \sin^4 \theta \cos^8 \theta \, d\theta = \frac{(3 \cdot 1) \cdot (7 \cdot 5 \cdot 3 \cdot 1)}{12 \cdot 10 \cdot 8 \cdot 6 \cdot 4 \cdot 2} \times \frac{\pi}{2} \] \[ = \frac{3 \times 105}{46080} \times \frac{\pi}{2} = \frac{315}{92160} \pi = \frac{7\pi}{2048} = \frac{7\pi}{2^{11}} \] Multiplying by the constant from
Step 2: \[ I = 2^{13} \times \frac{7\pi}{2^{11}} = 2^2 \times 7\pi = 4 \times 7\pi = 28\pi \]