1. Compute the following antiderivatives:

(a) $$\displaystyle{\int 2x^3\, dx}$$

(b) $$\displaystyle{\int 2x\sqrt{1+x^2}\, dx}$$

(c) $$\displaystyle{x^{50} \sin \big( x^{43} \big)}\, dx$$

(d) $$\displaystyle{\int x^3 \ln x\, dx}$$

(e) $$\displaystyle{\int e^x \sin x\, dx}$$

(f) $$\displaystyle{\int \sec x \tan x \, dx}$$

(g) $$\displaystyle{\int \frac{dx}{x^2+4}}$$

(h) $$\displaystyle{\int \frac{x^2}{\sqrt{25-x^2}}\, dx}$$

(i) $$\displaystyle{\int \frac{5}{(x+3)(x-2)}\, dx}$$

(j) $$\displaystyle{\int \frac{21x}{(x+3)(x+2)}\, dx}$$

(k) $$\displaystyle{\int \frac{x^3+5x^2+3x+11}{(x^2+4x+ 5)^2}\, dx}$$

(l) $$\displaystyle{\int \frac{x^4+7x^3-2}{x-11}\, dx}$$

2. Compute the derivative of the following functions:

(a) $$f(x) = \displaystyle{\int_3^{x^2} e^{t^4}\, dt}$$.

(b) $$f(x) = \displaystyle{\int_{\frac{x^3-2x^2}{\tan^2 x}}^{\sec^2 x} \cos^3 \big( t^5 \big) \, dt}$$.

3. Compute the following:

(a) $$\displaystyle{\int_5^\infty \sqrt{x+2}\, dx}$$.

(b) $$\displaystyle{\int_{-\infty}^\infty \frac{dx}{1+x^2} }$$.

(c) $$\displaystyle{\int_1^4 \frac{dx}{x-1} }$$.

(d) $$\displaystyle{\int_0^2 \frac{dx}{x(x-2)} }$$.

(e) $$\displaystyle{\int_0^{2\pi} \csc x\, dx}$$.

4. Find the area of the region bounded by the curves $$y=7x^2-2$$ and $$y=3-x/2$$.

5. Find the area of the region bounded by the curves $$y = \cos^2 x$$ and $$y = \sin^2 x$$ in the interval $$-\frac{\pi}{2} \leq x \leq \pi$$.

6. Find the volume of the solid obtained by rotating the graph of $$y = \sqrt{x}$$ between $$x=0$$ and $$x=4$$ around the x-axis.

7. Find the volume of the solid obtained by rotating the region bounded by the graphs of $$y = x$$ and $$y = \sqrt{x}$$ around the x-axis.

8. Find the volume of the solid obtained by rotating the region bounded by the graphs of $$x= 4y^{1/3}, x=0, x=10$$ around the y-axis.

9. Find the volume of the solid obtained by rotating the region bounded by the graphs of $$x=3, x=4+y^2/64, y=2, y=8$$, around the y-axis.

10. Find the volume of the solid obtained by rotating the region bounded by the graphs of $$y = 1/x, x=1, x=2$$, around the y-axis.

11. Find the volume of the solid obtained by rotating the region bounded by the graphs of $$x=y^3+3y^2, x=0$$, around the x-axis.

12. Find the volume of the solid obtained by rotating the region bounded by the graphs of $$x=1, x=4y^2-4y^3$$, around the line $$y=6$$.

13. Compute the next three terms of the following sequences:

(a) $$\big\{ 1, \dfrac{1}{3}, \dfrac{1}{5}, \dfrac{1}{9}, \dfrac{1}{11}, \dotsc, \big\}$$

(b) $$\big\{ 1, -\dfrac{3}{4}, \dfrac{9}{16}, -\dfrac{27}{64}, \dotsc, \big\}$$

14. Compute the following limits:

(a) $$\lim_{n\to\infty} \dfrac{n}{2n+1}$$

(b) $$\lim_{n\to\infty} \dfrac{5+3n^2}{n+3n^2}$$

15. Study the convergence of the following series:

(a) $$\displaystyle{\sum_{n=13}^\infty \sqrt{\frac{7n^3+5}{5n^5+3n^2+1}}}$$.

(b) $$\displaystyle{\sum_{n=1}^\infty \frac{5+6\sin(n^2)}{10^n}}$$.

(c) $$\displaystyle{\sum_{n=0}^\infty \frac{6}{5n^2+7n+3}}$$.

(d) $$\displaystyle{\sum_{n=2}^\infty \frac{2}{2^n-2}}$$.

(e) $$\displaystyle{\sum_{n=1}^\infty \frac{(-8)^n}{n!}}$$.

(f) $$\displaystyle{\sum_{n=1}^\infty \frac{(2n)!}{2^n n!}}$$.

(g) $$\displaystyle{\sum_{n=10}^\infty \frac{4n^3+5}{7n^2-11n^3}}$$.

(h) $$\displaystyle{\sum_{n=0}^\infty n^2 e^{-n}}$$.

(i) $$\displaystyle{\sum_{n=1}^\infty \frac{e^{2n}}{n^n}}$$.

(j) $$\displaystyle{\sum_{n=1}^\infty \frac{e^n \ln n}{3^n n^3}}$$.

(k) $$\displaystyle{\sum_{n=0}^\infty \frac{(-1)^{n-1}}{2n+1}}$$.

(l) $$\displaystyle{\sum_{n=0}^\infty \frac{\cos \pi n}{\sqrt{n}}}$$.

(m) $$\displaystyle{\sum_{n=0}^\infty \frac{(-1)^{n-1}}{2n+1}}$$.

(n) $$\displaystyle{\sum_{n=0}^\infty \frac{(-1)^{n-1}}{2n+1}}$$.

16. Find the sum of the following series:

(a) $$\displaystyle{\sum_{n=1}^\infty 4682 \big( \tfrac{1}{2} \big)^n}$$.

(b) $$\displaystyle{\sum_{n=0}^\infty 47 \big( \tfrac{5}{64} \big)^{n+3}}$$.

(c) $$\displaystyle{\sum_{n=1}^\infty \frac{1}{4n^2-1}}$$.

17. Find a power series representation of the following functions:

(a) $$f(x) = \dfrac{1}{4+x^2}$$.

(b) $$f(x) = \dfrac{5}{2-3x}$$.

(c) $$f(x) = \dfrac{3}{(x-1)(x+2)}$$.

(d) $$f(x) = \dfrac{\ln (3x)}{(x^2-2)(x+5)}$$.

18. Find the Taylor series of the following functions:

(a) $$f(x) = 10^x$$ at $$x = 0$$.

(b) $$f(x) = \cos(10 x)$$ at $$x = 0$$.

19. Find the center and radius of convergence of the following series

(a) $$\displaystyle{\sum_{n=0}^\infty (-3)^n \sqrt{n+1} (x+1)^n }$$.

(b) $$\displaystyle{\sum_{n=0}^\infty \frac{(-1)^n}{n 10^n} (x-2)^n }$$.

20. Find the interval of convergence of the following series:

(a) $$\displaystyle{\sum_{n=0}^\infty \frac{x^n}{n!}}$$.

21. Find a power series representation for the function $$f(x) = \dfrac{\ln (x-5)}{(x+10)}$$, and compute center, radius of convergence, and domain.

22. Compute the area of one loop of the curve given in polar coordinates by $$r = \sin (5\theta)$$.

23. Compute the area of the region bounded by the graphs of the curves given in polar coordinates by $$r = 3 \cos \theta$$ and $$r = 3 \sin \theta$$.

24. Express in rectangular coordinates the curves given in polar coordinates by $$r=2$$ and $$r=1-\sin\theta$$.

25. Express in polar coordinates the curves given in rectangular coordinates by $$x+y=5$$ and $$7y=x^2$$.

26. Find the length of the following polar curves:

(a) $$r=4$$ for $$0 \leq \theta \leq 13\pi/20$$.

(b) $$r = e^{5\theta}$$ for $$0 \leq \theta \leq \pi$$.

27. Sketch the curve given in parametric equations by $$x = 2\sin t, y = \cos^2 t$$ for $$0 \leq t \leq 2\pi$$.

28. Sketch the curve given in parametric equations by $$x = 3\cos(5t), y = 1 + \cos(5t)$$ for $$0 \leq t \leq \pi/2$$.