- Demonstrate how to evaluate the following integrals by parts. \[ \int x \sin(x) \,\mathrm{d}x \qquad \int t \mathrm{e}^{3t} \,\mathrm{d}t \qquad \int t^2 \mathrm{e}^t \,\mathrm{d}t \] \[ \int\limits \mathrm{arctan}(x) \,\mathrm{d}x \; {\color{gray}{\Big(\text{Hint: it's like\(\int\ln(x)\)} \Big)}} \qquad \int \mathrm{e}^{2x}\cos(x) \,\mathrm{d}x \] \[ \int x^2 2^x \,\mathrm{d}x \qquad \int x \ln(x) \,\mathrm{d}x \qquad \int \Big(\ln(x)\Big)^2 \,\mathrm{d}x \]
- What are the exact values of these definite integrals? \[ \int\limits_0^1 \arctan(x) \,\mathrm{d}x \qquad \int\limits_{1}^{\mathrm{e}} t^2 \mathrm{e}^t \,\mathrm{d}t \qquad \int\limits_{1}^{\mathrm{e}} x \ln(x) \,\mathrm{d}x \]