## Review for the Final Exam

Feel free to comment below if you need some guidance with any problem. As it is customary, I will provide with hints, but no solutions.

1. Evaluate the integral $\displaystyle{\int t^2 e^t\, dt}$
2. Evaluate the integral $\displaystyle{\int (x-3) \sqrt{ x^2-6x+5 }\, dx}$
3. Evaluate the integral $\displaystyle{\int \frac{1}{x^3 e^{1/x}}\, dx}$
4. Evaluate the integral $\displaystyle{\int \frac{x^3+1}{(x+1)^2(x^2+4)}\, dx}$
5. Evaluate the following integral, or indicate if it is divergent:
$\displaystyle{\int_0^\infty \frac{x \tan^{-1}x}{(1+x^2)^{3/2}}\, dx}$
6. Find the volume of the solid obtained by rotating the region bounded by $y=x^2$ and $y=2-x$ around the line $x=1.$
7. Find the volume of the solid obtained by rotating the region bounded by $y=e^{-x}, y=1/e,$ and $x=0$ around the line $y=0.$
8. Find the general term of the sequence $\big\{ 3,2, \frac{5}{3}, \frac{3}{2}, \frac{7}{5}, \frac{4}{3}, \dotsc \big\},$ and compute its limit.
9. Compute the limit of the sequence
$\bigg\{ \displaystyle{\frac{n^2+5n+2}{\sqrt{n^4+1}}} \bigg\}_{n=1}^\infty$
10. Compute the limit of the sequence $\big\{ \tan (\pi - 1/n) \big\}_{n=1}^\infty$
11. Compute $\displaystyle{ \lim_{n\to \infty} \bigg( 1- \frac{2}{n} \bigg)^n }$
12. Study the convergence of the series $\displaystyle{\sum_{n=2}^\infty \frac{3^n+4^n}{5^n}}.$ If convergent, evaluate the sum.
13. Classify the series $\displaystyle{\sum_{n=1}^\infty \frac{\cos (\pi n)}{n^{2/3}}}$ as absolutely convergent, conditionally convergent, or divergent.
14. Classify the series $\displaystyle{\sum_{n=1}^\infty \frac{(-1)^n n}{e^n}}$ as absolutely convergent, conditionally convergent, or divergent.
15. Find the interval of convergence of the power series $\displaystyle{ \sum_{n=0}^\infty \frac{(-3)^n x^n}{\sqrt{n+1}}}.$
16. Express the function $f(x) = \displaystyle{\frac{2x}{x^3+8}}$ as a power series.