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 \( 0 \leq t \leq \pi/2 \).