Question

\(\int_{-1}^{1} \sin^7 x \cos^{13} x \, dx = ? \)

• A. 0
• B. 1
• C. 20
• D. 6

Hint:

When integrating odd functions over symmetric intervals, remember that the result will always be zero.

Answer 1

The Correct Option is A

We are asked to evaluate the integral: \[ \int_{-1}^{1} \sin^7 x \cos^{13} x \, dx \] Step 1: Analyze the integrand Notice that the integrand contains powers of \(\sin x\) and \(\cos x\). Specifically, \(\sin^7 x\) is an odd function, and \(\cos^{13} x\) is an odd function raised to an odd power. Step 2: Properties of odd functions The product of an odd function and an even function is an odd function. Since \(\sin^7 x\) is odd and \(\cos^{13} x\) is even, the integrand is an odd function. Step 3: Integral of an odd function The integral of an odd function over a symmetric interval \([-a, a]\) is always 0. Therefore: \[ \int_{-1}^{1} \sin^7 x \cos^{13} x \, dx = 0 \] Thus, the correct answer is: (A) 0.