Full solution

\int \sin^{2}x \cos^{2}x \, dx

Recognize the integral: Recognize the integral that needs to be solved. $\newline$ We need to integrate the function $\sin^2(x)\cos^2(x)$ with respect to $x$ .
Use trigonometric identities: Use trigonometric identities to simplify the integrand. $\newline$ We can use the double angle identity for cosine, which is $\cos(2x) = 2\cos^2(x) - 1$ or $\cos^2(x) = \frac{1 + \cos(2x)}{2}$ . Similarly, for sine, we have $\sin^2(x) = \frac{1 - \cos(2x)}{2}$ .
Substitute the identities: Substitute the identities into the integral. The integral becomes $\int\left[\frac{1 - \cos(2x)}{2}\right] * \left[\frac{1 + \cos(2x)}{2}\right] dx$ .
Expand the integrand: Expand the integrand. $\newline$ Expanding the integrand gives us $\int(\frac{1}{4} - \frac{\cos^2(2x)}{4}) dx$ .
Split the integral: Split the integral into two separate integrals. $\newline$ We now have $(\frac{1}{4})\int dx - (\frac{1}{4})\int \cos^2(2x) dx$ .
Integrate the first part: Integrate the first part of the integral. $\newline$ The integral of $dx$ is $x$ , so the first part becomes $(1/4)x$ .
Use a trigonometric identity: Use a trigonometric identity to simplify the second integral. $\newline$ We can use the power reduction identity for $\cos^2(2x)$ , which is $\cos^2(2x) = \frac{1 + \cos(4x)}{2}$ .
Substitute the identity: Substitute the identity into the second integral. $\newline$ The second integral becomes $(\frac{1}{4})\int(\frac{1}{2} + \frac{\cos(4x)}{2}) dx$ .
Simplify the second integral: Simplify the second integral. $\newline$ This simplifies to $(\frac{1}{8})\int dx + (\frac{1}{8})\int \cos(4x) dx$ .
Integrate the simplified second integral: Integrate the simplified second integral. $\newline$ The integral of $dx$ is $x$ , and the integral of $\cos(4x)$ is $(1/4)\sin(4x)$ , so the second part becomes $(1/8)x + (1/32)\sin(4x)$ .
Combine the results: Combine the results from Step $6$ and Step $10$ . $\newline$ The final answer is $(\frac{1}{4})x - [\left(\frac{1}{8}\right)x + \left(\frac{1}{32}\right)\sin(4x)] + C$ , where $C$ is the constant of integration.
Simplify the final expression: Simplify the final expression. $\newline$ Combine like terms to get the final answer: $\frac{1}{8}x - \frac{1}{32}\sin(4x) + C$ .