Homework 9: Calculus and Analytic Geometry

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 24 February 2010

Problem A. Indefinite integrals.

1. $\int x^7\mathrm{dx}$
2. $\int x^{-7}\mathrm{dx}$
3. $\int x^{-1}\mathrm{dx}$
4. $\int x^{\frac{5}{3}}\mathrm{dx}$
5. $\int x^{-\frac{5}{4}}\mathrm{dx}$
6. $\int \sqrt[3]{x^2}\mathrm{dx}$
7. $\int \frac{1}{\sqrt[4]{x^3}}\mathrm{dx}$
8. $\int \frac{2}{x^2}\mathrm{dx}$
9. $\int \left(8-x+2x^3-\frac{6}{x^3}+2x^{-5}+5x^{-1}\right)\mathrm{dx}$
10. $\int \left(2-5x\right)\left(3+2x\right)\left(1-x\right)\mathrm{dx}$
11. $\sqrt{x}\left(a{x}^2+bx+c\right)\mathrm{dx}$
12. $\int {\left(x^2-\frac{1}{x^2}\right)}^3\mathrm{dx}$
13. $\int \left(\sqrt{x}-\frac{1}{\sqrt{x}}\right)\mathrm{dx}$
14. $\int {\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)}^2\mathrm{dx}$
15. $\int \frac{{\left(1+2x\right)}^3}{x^4}\mathrm{dx}$
16. $\int \frac{{\left(1+x\right)}^3}{\sqrt{x}}\mathrm{dx}$
17. $\frac{2x^2+x-2}{x-2}\mathrm{dx}$
18. If $\frac{\mathrm{df}}{\mathrm{dx}}=x-\frac{1}{x^2}$ and $f\left(1\right)=\frac{1}{2}$ find $f\left(x\right).$

Problem B. Indefinite Integrals with Trigonometric Functions.

1. $\int \left(9\sin x-7\cos x-\frac{6}{{\cos }^{2}x}+\frac{2}{{\sin }^{2}x}+{\cot }^{2}x\right)\mathrm{dx}$
2. $\int \left(\frac{\cot x}{\sin x}-{\tan }^{2}x-\frac{\tan x}{\cos x}+\frac{2}{{\cos }^{2}x}\right)\mathrm{dx}$
3. $\int \sec x\left(\sec x+\tan x\right)\mathrm{dx}$
4. $\int \csc x\left(\csc x-\cot x\right)\mathrm{dx}$
5. $\int {\left(\tan x+\cot x\right)}^2\mathrm{dx}$
6. $\int \frac{1+2\sin x}{{\cos }^{2}x}\mathrm{dx}$
7. $\int \frac{3\cos x+4}{{\sin }^{2}x}\mathrm{dx}$
8. $\int \frac{1}{1-\cos x}\mathrm{dx}$
9. $\int \frac{1}{1+\cos x}\mathrm{dx}$
10. $\frac{\tan x}{\sec x+\tan x}\mathrm{dx}$
11. $\int \frac{\csc x}{\csc x-\cot x}\mathrm{dx}$
12. $\int \frac{\cos x}{1+\cos x}\mathrm{dx}$
13. $\int \frac{\sin x}{1-\sin x}\mathrm{dx}$
14. $\int \sqrt{1+\cos 2x}\mathrm{dx}$
15. $\int \sqrt{1-\cos 2x}\mathrm{dx}$
16. $\int \frac{1}{1+\cos 2x}\mathrm{dx}$
17. $\int \frac{1}{1-\cos 2x}\mathrm{dx}$
18. $\int \sqrt{1+\sin 2x}\mathrm{dx}$
19. $\int \frac{{\sin }^{3}x+{\cos }^{3}x}{{\sin }^{2}x{\cos }^{2}x}\mathrm{dx}$

Problem C. Integrals with exponential functions and inverse functions.

1. $\int 2^x\mathrm{dx}$
2. $\int 6x^5-2x^{-4}-7x+\frac{3}{x}-5+4e^x+7^x\mathrm{dx}$
3. $\int \left(\frac{x}{a}+\frac{a}{x}+x^a+a^x\right)\mathrm{dx}$
4. $\left(\sqrt{x}-\sqrt[3]{x^4}+\frac{7}{\sqrt[3]{x^2}}-6e^x+1\right)\mathrm{dx}$
5. $\int \frac{x^2-1}{x^2+1}\mathrm{dx}$
6. $\int \frac{x^6-1}{x^2+1}\mathrm{dx}$
7. $\int \frac{x^4}{1+x^2}\mathrm{dx}$
8. $\int \frac{x^2}{1+x^2}\mathrm{dx}$
9. $\int \left(1+\frac{1}{1+x^2}-\frac{2}{\sqrt{1-x^2}}+\frac{5}{x\sqrt{x^2-1}}+a^x\right)\mathrm{dx}$
10. $\int {\tan }^{-1}\left(\frac{\sin 2x}{1+\cos 2x}\right)\mathrm{dx}$
11. $\int {\cos }^{-1}\left(\frac{1-{\tan }^{2}x}{1+{\tan }^{2}x}\right)\mathrm{dx}$
12. $\int {\cos }^{-1}\left(\sin x\right)\mathrm{dx}$
13. $\int {\cot }^{-1}\left(\frac{\sin 2x}{1-\cos 2x}\right)\mathrm{dx}$

Problem D. Integration by substitution.

1. $\int {\left(2x+9\right)}^5\mathrm{dx}$
2. $\int {\left(7x-3\right)}^4\mathrm{dx}$
3. $\int \sqrt{3x-5}\mathrm{dx}$
4. $\int \frac{1}{\sqrt{4x+3}}\mathrm{dx}$
5. $\int \frac{1}{\sqrt{3x-4}}\mathrm{dx}$
6. $\int \frac{1}{{2x-3}^{\frac{3}{2}}}\mathrm{dx}$
7. $\int \frac{4x}{2x^2+3}\mathrm{dx}$
8. $\frac{x+1}{x^2+2x-3}\mathrm{dx}$
9. $\int \frac{4x-5}{2x^2-5x+1}\mathrm{dx}$
10. $\int \frac{9x^2-4x+5}{3x^3-2x^2+5x+1}\mathrm{dx}$
11. $\int \frac{2x+3}{\sqrt{x^2+3x-2}}\mathrm{dx}$
12. $\int \frac{2x-1}{\sqrt{x^2-x-1}}\mathrm{dx}$
13. $\int \frac{\mathrm{dx}}{\sqrt{x+a}+\sqrt{x+b}}$
14. $\int \frac{\mathrm{dx}}{\sqrt{1-3x}-\sqrt{5-3x}}$
15. $\int \frac{x^2}{1+x^6}\mathrm{dx}$
16. $\int \frac{x^3}{1+x^8}\mathrm{dx}$
17. $\int \frac{x}{1+x^4}\mathrm{dx}$
18. $\int \frac{x^5}{\sqrt{1+x^3}}\mathrm{dx}$
19. $\int \frac{x}{\sqrt{1+x}}\mathrm{dx}$
20. $\int \frac{1}{x\sqrt{x^4-1}}\mathrm{dx}$
21. $\int x\sqrt{x-1}\mathrm{dx}$
22. $\int \left(1-x\right)\sqrt{1+x}\mathrm{dx}$
23. $\int x\sqrt{x^2-1}\mathrm{dx}$
24. $\int x\sqrt{3x-2}\mathrm{dx}$
25. $\int \left(2x-3\right)\sqrt{x^2-3x+5}\mathrm{dx}$
26. $\int \frac{\mathrm{dx}}{3-5x}$
27. $\int \sqrt{1+x}\mathrm{dx}$

Problem E. Integrals with trigonometric functions

1. $\int \sin 3x\mathrm{dx}$
2. $\int \cos \left(5+6x\right)\mathrm{dx}$
3. $\int \sin \left(5-3x\right)\mathrm{dx}$
4. $\int {\csc }^2\left(2x+5\right)\mathrm{dx}$
5. $\int \sin x\cos x\mathrm{dx}$
6. $\int {\sin }^{3}x\cos x\mathrm{dx}$
7. $\int \sqrt{\cos x}\sin x\mathrm{dx}$
8. $\int \frac{\cos \sqrt{x}}{\sqrt{x}}\mathrm{dx}$
9. $\int \sin \left(ax+b\right)\cos \left(ax+b\right)\mathrm{dx}$
10. $\int {\cos }^{3}x\mathrm{dx}$
11. $\int \left(\frac{1}{x^2}\right)\cos \left(\frac{1}{x}\right)\mathrm{dx}$
12. $\int 2x\sin \left(x^2+1\right)\mathrm{dx}$
13. $\int \frac{\tan \sqrt{x}{\sec }^2\sqrt{x}}{\sqrt{x}}\mathrm{dx}$
14. $\int \frac{{\sec }^{2}x}{1+\tan x}\mathrm{dx}$
15. $\int \frac{\sin x}{1+\cos x}\mathrm{dx}$
16. $\int \frac{\sin x}{2+3\cos x}\mathrm{dx}$
17. $\int \frac{\sin 2x}{a^2+b^2{\sin }^{2}x}\mathrm{dx}$
18. $\int \frac{\sin 2x}{a^2{\cos }^{2}x+b^2{\sin }^{2}x}\mathrm{dx}$
19. $\int \frac{2\cos x-3\sin x}{3\cos x+2\sin x}\mathrm{dx}$
20. $\int \frac{1+\cos x}{{\left(\mathrm{x}+\sin x\right)}^3}\mathrm{dx}$
21. $\int \frac{\sin x}{{\left(1+\cos x\right)}^2}\mathrm{dx}$
22. $\int x^2\sin x^3\mathrm{dx}$
23. $\int \frac{\sin x}{\sin x-\cos x}\mathrm{dx}$
24. $\int \frac{\mathrm{dx}}{1-\tan x}$
25. $\int \frac{\mathrm{dx}}{1-\cot x}$
26. $\int \frac{\cos 2x}{{\left(\sin x+\cos x\right)}^2}\mathrm{dx}$
27. $\int \frac{\cos x-\sin x}{\left(1+\sin 2x\right)}\mathrm{dx}$
28. $\int x{\sin }^3{x}^2\cos x^2\mathrm{dx}$
29. $\int \frac{{\sec }^{2}x}{\sqrt{1-{\tan }^{2}x}}\mathrm{dx}$
30. $\int 2x{\sec }^3\left(x^2+3\right)\tan \left(x^2+3\right)\mathrm{dx}$
31. $\int \frac{\sin 2x}{{\left(a+b\cos 2x\right)}^2}\mathrm{dx}$

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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