1. **Stating the problem:** You want to integrate a function that is squared, but you cannot or do not want to take the square root. 2. **General approach:** When integrating a squared function $f(x)^2$, you cannot simply take the square root to simplify. Instead, you use integration techniques such as substitution, integration by parts, or trigonometric identities depending on the function. 3. **Formula and rules:** The integral of $f(x)^2$ is generally written as $$\int f(x)^2 \, dx.$$ There is no direct formula for all functions squared, but some useful identities include: - For trigonometric functions: $$\sin^2(x) = \frac{1 - \cos(2x)}{2}, \quad \cos^2(x) = \frac{1 + \cos(2x)}{2}.$$ - For polynomials or other functions, consider substitution or expanding the square. 4. **Example:** Suppose you want to integrate $$\int (x^2 + 1)^2 \, dx.$$ 5. **Expand the square:** $$ (x^2 + 1)^2 = x^4 + 2x^2 + 1. $$ 6. **Rewrite the integral:** $$ \int (x^2 + 1)^2 \, dx = \int (x^4 + 2x^2 + 1) \, dx. $$ 7. **Integrate term-by-term:** $$ \int x^4 \, dx = \frac{x^5}{5}, \quad \int 2x^2 \, dx = 2 \cdot \frac{x^3}{3} = \frac{2x^3}{3}, \quad \int 1 \, dx = x. $$ 8. **Combine results:** $$ \int (x^2 + 1)^2 \, dx = \frac{x^5}{5} + \frac{2x^3}{3} + x + C, $$ where $C$ is the constant of integration. 9. **Summary:** When you cannot take the square root, try expanding the square or use identities and integration techniques appropriate to the function type.