1. **State the problem:** We need to evaluate the integral $$\int x^2 \sec^2(x) \, dx$$. 2. **Recall the formula and rules:** The integral involves a product of $x^2$ and $\sec^2(x)$. We can use integration by parts, which states: $$\int u \, dv = uv - \int v \, du$$ Choose $u = x^2$ (which simplifies when differentiated) and $dv = \sec^2(x) dx$ (which integrates easily). 3. **Compute derivatives and integrals:** - $du = 2x \, dx$ - $v = \tan(x)$ since $\frac{d}{dx} \tan(x) = \sec^2(x)$ 4. **Apply integration by parts:** $$\int x^2 \sec^2(x) \, dx = x^2 \tan(x) - \int \tan(x) (2x) \, dx = x^2 \tan(x) - 2 \int x \tan(x) \, dx$$ 5. **Evaluate the remaining integral $\int x \tan(x) \, dx$ using integration by parts again:** - Let $u = x$, so $du = dx$ - Let $dv = \tan(x) dx$, so $v = -\ln|\cos(x)|$ 6. **Apply integration by parts again:** $$\int x \tan(x) \, dx = -x \ln|\cos(x)| + \int \ln|\cos(x)| \, dx$$ 7. **Combine all parts:** $$\int x^2 \sec^2(x) \, dx = x^2 \tan(x) - 2 \left(-x \ln|\cos(x)| + \int \ln|\cos(x)| \, dx \right) + C$$ 8. **Final answer:** $$\boxed{\int x^2 \sec^2(x) \, dx = x^2 \tan(x) + 2x \ln|\cos(x)| - 2 \int \ln|\cos(x)| \, dx + C}$$ Note: The integral $\int \ln|\cos(x)| \, dx$ does not have a simple elementary form and is typically left as is or expressed in terms of special functions.