Question

Evaluate the integral: \[ \int_{-2}^{2} (4 - x^2)^{\frac{5}{2}} \, dx. \]

• A. \( 40\pi \)
• B. \( 20\pi \)
• C. \( 10\pi \)
• D. \( \frac{5\pi}{32} \)

Hint:

For integrals of the form \( \int_{-a}^{a} (a^2 - x^2)^n dx \), consider using trigonometric substitution \( x = a\sin\theta \) to simplify the integral.

Answer 1

The Correct Option is B

Step 1: Recognizing the Integral Type
The given integral has the standard form: \[ I = \int_{-a}^{a} (a^2 - x^2)^n \, dx. \] which suggests using a trigonometric substitution.
Step 2: Substituting \( x = 2\sin\theta \) Let: \[ x = 2\sin\theta, \quad dx = 2\cos\theta d\theta. \] Rewriting the integral: \[ I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (4 - 4\sin^2\theta)^{\frac{5}{2}} \cdot 2\cos\theta \, d\theta. \] Using \( \cos^2\theta = 1 - \sin^2\theta \), we get: \[ (4 - 4\sin^2\theta)^{\frac{5}{2}} = 4^{\frac{5}{2}} \cos^{5} \theta. \] Step 3: Evaluating the Integral \[ I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} 32 \cos^6\theta \cdot 2\cos\theta \, d\theta. \] \[ I = 64 \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos^6\theta \, d\theta. \] Using the reduction formula: \[ \int_{0}^{\frac{\pi}{2}} \cos^{2n} \theta \, d\theta = \frac{\pi}{2} \frac{(2n-1)!!}{(2n)!!}. \] For \( n = 3 \): \[ \int_{0}^{\frac{\pi}{2}} \cos^6\theta \, d\theta = \frac{\pi}{16} \frac{15}{16}. \] Multiplying by 64: \[ I = 64 \times \frac{5\pi}{16} = 20\pi. \] Step 4: Conclusion
Thus, the correct answer is: \[ \mathbf{20\pi}. \]