Question

Evaluate \[ \int_{-1}^{1} \sin^5x \cos^4x \, dx \]

Hint:

When integrating even and odd functions over symmetric limits, you can often simplify the calculation by recognizing symmetry in the integrand.

Answer 1

Step 1: Understand the integral.
The given integral is: \[ \int_{-1}^{1} \sin^5x \cos^4x \, dx. \] This is an even function because the powers of \( \sin x \) and \( \cos x \) are both odd and even, respectively. Therefore, we can evaluate the integral over \( [0, 1] \) and then multiply by 2. \[ \int_{-1}^{1} \sin^5x \cos^4x \, dx = 2 \int_{0}^{1} \sin^5x \cos^4x \, dx. \]

Step 2: Solution.
The exact evaluation would require applying standard methods, possibly substitution or known reduction formulas, but it's clear from symmetry and integral properties that the result is zero (since the function is odd over symmetric limits).

Step 3: Conclusion.
Thus, the value of the integral is: \[ \boxed{0}. \]