1. **State the problem:** We need to evaluate the integral $$\int t^2 \cot t \, dt$$. 2. **Recall the formula and rules:** The cotangent function is $$\cot t = \frac{\cos t}{\sin t}$$. Integration by parts is useful here, where $$\int u \, dv = uv - \int v \, du$$. 3. **Choose parts:** Let $$u = t^2$$ and $$dv = \cot t \, dt$$. 4. **Compute derivatives and integrals:** - $$du = 2t \, dt$$ - To find $$v$$, integrate $$dv = \cot t \, dt$$: $$v = \int \cot t \, dt = \ln|\sin t| + C$$ 5. **Apply integration by parts:** $$\int t^2 \cot t \, dt = t^2 \ln|\sin t| - \int \ln|\sin t| \cdot 2t \, dt$$ 6. **Simplify:** $$= t^2 \ln|\sin t| - 2 \int t \ln|\sin t| \, dt$$ 7. **The remaining integral $$\int t \ln|\sin t| \, dt$$ is more complex and typically requires advanced techniques or special functions.** **Final answer:** $$\int t^2 \cot t \, dt = t^2 \ln|\sin t| - 2 \int t \ln|\sin t| \, dt + C$$ This expresses the integral in terms of a simpler integral that may be evaluated numerically or with further methods.