Question

\( \int_{-4\pi}^{4\pi} \tan^9 x \sin^6 x \cos^3 x \, dx = \)

• A. \( 16 \times \frac{\pi}{2} \)
• B. \( 8 \times \frac{2}{3} \)
• C. \( 16 \times \frac{14}{17} \times \frac{12}{15} \times \cdots \times \frac{2}{3} \)
• D. \( 0 \)

Hint:

Recognizing odd and even functions can greatly simplify definite integrals over symmetric intervals.

Answer 1

The Correct Option is D

Step 1: Check if the integrand is odd or even.
Let \( f(x) = \tan^9 x \sin^6 x \cos^3 x \). \( f(-x) = (-\tan x)^9 (-\sin x)^6 (\cos x)^3 = -\tan^9 x \sin^6 x \cos^3 x = -f(x) \). So, \( f(x) \) is an odd function.
Step 2: Apply the property of definite integrals for odd functions.
For an odd function \( f(x) \), \( \int_{-a}^{a} f(x) \, dx = 0 \). Here, \( a = 4\pi \), so \( \int_{-4\pi}^{4\pi} \tan^9 x \sin^6 x \cos^3 x \, dx = 0 \).