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.
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