1. **State the problem:** We need to find the integral $\int \ln x^2 \, dx$. 2. **Simplify the integrand:** Using the logarithm power rule, $\ln x^2 = 2 \ln x$. 3. **Rewrite the integral:** $$\int \ln x^2 \, dx = \int 2 \ln x \, dx = 2 \int \ln x \, dx$$ 4. **Recall the formula for $\int \ln x \, dx$:** Use integration by parts where $u = \ln x$ and $dv = dx$. 5. **Integration by parts:** $$du = \frac{1}{x} dx, \quad v = x$$ 6. **Apply integration by parts formula:** $$\int u \, dv = uv - \int v \, du$$ 7. **Calculate:** $$\int \ln x \, dx = x \ln x - \int x \cdot \frac{1}{x} dx = x \ln x - \int 1 \, dx = x \ln x - x + C$$ 8. **Substitute back:** $$2 \int \ln x \, dx = 2 (x \ln x - x) + C = 2x \ln x - 2x + C$$ **Final answer:** $$\int \ln x^2 \, dx = 2x \ln x - 2x + C$$