Consider \( \int \sin ^{4}(a \boldsymbol{x}) d x \) Expansion of \( \sin ^{4}(a x) \) gives us \( \begin{array}{l}(3 / 8)-(1 / 2) \cos a x+(1 / 8) \cos 4 a x \\ (1 / 8)+(1 / 2) \cos a x+(1 / 8) \cos 4 a x \\ (1 / 8)+(1 / 2) \cos a x-(1 / 8) \cos 4 a x \\ (3 / 8)+(1 / 2) \cos a x+(1 / 8) \cos 4 a x \\ (1 / 8)+(3 / 2) \cos a x+(1 / 8) \cos 4 a x\end{array} \)

Let's break down \( \sin^4(a x) \). Using the power-reduction formula, we find that we can express \( \sin^4(a x) \) in terms of cosine. The correct expansion is: \[ \sin^4(a x) = \left(\sin^2(a x)\right)^2 = \left(\frac{1 - \cos(2a x)}{2}\right)^2 = \frac{1}{4}(1 - 2\cos(2a x) + \cos^2(2a x)) \] Now, using the power-reduction formula again on \( \cos^2(2a x) \), we get: \[ \cos^2(2a x) = \frac{1 + \cos(4a x)}{2} \] Putting everything together, the expansion becomes: \[ \sin^4(a x) = \frac{1}{4} \left( 1 - 2\cos(2a x) + \frac{1 + \cos(4a x)}{2} \right) = \frac{3}{8} - \frac{1}{2} \cos(2a x) + \frac{1}{8} \cos(4a x) \] Thus, the correct expression for \( \sin^4(a x) \) is: \[ \sin^4(a x) = \frac{3}{8} - \frac{1}{2} \cos(2a x) + \frac{1}{8} \cos(4a x) \] Keep this formula in mind, as it greatly simplifies evaluating the integral! Now, let's consider what this integral represents and where you might see it in the real world. The integral of \(\sin^4(ax)\) has useful applications in physics, particularly in wave mechanics and quantum mechanics, where fluctuations are analyzed. Specifically, the integral can represent the energy contained within certain frequencies in waves when looking at their time evolution. When approaching integrals like this, remember to always simplify the trigonometric functions using identities. A common mistake is to miss extra factors from the identities, which can lead to incorrect results. Double-check each step for correctness!