To solve the integral \(\int_{-\pi/4}^{\pi/4} \sin^{103} x \cos^{101} x \, dx\), we need to analyze the symmetry properties of the integrand.
The given integral is:
\(I = \int_{-\pi/4}^{\pi/4} \sin^{103} x \cos^{101} x \, dx\)
Notice that \(\sin^{103} x\) is an odd function, and \(\cos^{101} x\) is an odd function because any power of cosine for odd numbers retains the odd nature. Thus, their product, \(\sin^{103} x \cos^{101} x\), is an odd function.
An odd function is defined by the property \(f(-x) = -f(x)\).
The integral of an odd function over a symmetric interval \([-a, a]\) is zero:
\(\int_{-a}^{a} f(x) \, dx = 0\) for odd \(f(x)\).
Therefore, since we are integrating an odd function over the symmetric interval \([-π/4, π/4]\), the integral evaluates to:
\(I = 0\)
Hence, the correct answer is 0.