Week 6 Exercises

6.3 Sheet 13 Integration by substitution

By using suitable substitution, integrate the following

1. \(\int (x+4)^2\ dx\)
2. \(\int \sqrt{(2x+3)}\ dx\)
3. \(\int (2-x)^3\ dx\)
4. \(\int e^{3x}\ dx\)
5. \(\int 1/(2x+1)\ dx\)
6. \(\int 1/(2x+1)^3\ dx\)
7. \(\int (x^2+1)^52x \ dx\)
8. \(\int (x^2+1)^3\ dx\)
9. \(\int \sqrt{(x^2+x)}(2x+1) \ dx\)
10. \(\int (ax^2+bx)^4(2ax+b)\ dx\)
11. \(\int (2x+1)\cos(x^2+x) \ dx\)
12. \(\int e^{5x^2-x}(10x-1)\ dx\)
13. \(\int \ln(x)/x \ dx\)
14. \(\int e^x/(1+e^x)\ dx\)
15. \(\int \cos(x/2+1)\ dx\)
16. \(\int 1/(x-1)^2\ dx\)
17. \(\int_0^1 (2x-1)^7\ dx\)
18. \(\int_3^5 3x\sqrt{x^2-9}\ dx\)
19. \(\int\sin(6x)\cos(5x) \ dx\)
20. \(\int \cos(2x)\sin(5x)\ dx\)
21. \(\int \cos^4(\theta)\sin(\theta)\ d\theta\)
22. \(\int \cos(\theta)(\sin^4(\theta)+2)\ d\theta\)
23. \(\int \cos(x)/\sin(x)\ dx\)
24. \(\int \sin(\theta)/\cos(\theta)\ d\theta\)
25. \(\int 1/(1+x^2)\ dx\)
26. \(\int 1/\sqrt{4-x^2}\ dx\)
27. \(\int \tan(\theta)\sec^2(\theta)\ d\theta\)
28. \(\int (6x+2)(3x^2+2x-9)\ dx\)
29. \(\int 4x^3/(x^4-1)\ dx\)
30. \(\int \sec^2(x)/\tan(x)\ dx\)
31. \(\int 1/\sqrt{2-x^2}\ dx\)
32. \(\int 1/\sqrt{9-2x^2}\ dx\)
33. \(\int 1/(5+4x^2)\ dx\)
34. \(\int \sqrt{4-x^2}\ dx\)

{Solutions: 1. \(\frac{1}{3}(x+4)^3+C\); 2. \(\frac{1}{3}(2x+3)^{3/2}+C\); 3. \(-\frac{1}{4}(2-x)^4 +C\); 4. \(\frac{1}{3}\exp(3x)+C\); 5. \(\frac{1}{2}\ln(|2*x+1|)+C\); 6. \(-\frac{1}{4}(2x+1)^{-2}+C\); 7. \(\frac{1}{6}(x^2+1)^6+C\); 8. \(\frac{1}{7}x^7+\frac{3}{5}x^5+x^3+x+C\); 9. \(\frac{2}{3}(x^2+x)^{3/2}+C\); 10. \(\frac{1}{5}(ax^2+bx)^{5}+C\); 11. \(\sin(x^2+x)+C\); 12. \(\exp(5x^2-x)+C\); 13. \(\frac{1}{2}\ln^2(x)+C\); 14. \(\ln(1+e^x)+C\); 15. \(2\sin(x/2+1)+C\); 16. \(-\frac{1}{x-1}+C\); 17. \(0\); 18. \(64\); 19. \(-\frac{1}{22}\cos(11x)-\frac{1}{2}\cos(x)+C\); 20. \(-\frac{1}{14}\cos(7x)-\frac{1}{6}\cos(3x)+C\); 21. \(-\frac{1}{5}\cos^5(x)+C\); 22. \(\frac{1}{5}\sin^5(\theta)+2\sin(\theta)+C\); 23. \(\ln(|\sin(x)|)+C\); 24. \(-\ln(|\cos(\theta)|)+C\); 25. \(\arctan(x)+C\); 26. \(\arcsin(x/2)+C\); 27. \(\frac{1}{2}\tan^2(\theta)+C\); 28. \(\frac{1}{2}(3x^2+2x-9)^2+C\); 29. \(\ln(|x^4-1|)+C\); 30. \(\ln(|\tan(x)|)+C\); 31. \(\arcsin(x/\sqrt{2})+C\); 32. \(\frac{1}{\sqrt{2}}\arcsin(\sqrt{2}x/3) +C\); 33. \(\frac{1}{2\sqrt{5}}\arctan(2x/\sqrt{5}) +C\); 34. \(2\arcsin(x/2) + \sin\left(2\arcsin(x/2)\right) +C = 2\arcsin(x/2)+\frac{x}{2}\sqrt{4-x^2}\); }

6.4 Sheet 14 - Integration by parts

Integrate the following integrals using integration by parts

1. \(\int x\cos(x)\ dx\)
2. \(\int x\ln(x)\ dx\)
3. \(\int x\sin(4x)\ dx\)
4. \(\int x(1+x)^{10}\ dx\)
5. \(\int x^7\ln(x)\ dx\)
6. \(\int \sqrt{x}\ln(x)\ dx\)
7. \(\int x^2e^x\ dx\)
8. \(\int x^2\cos(x)\ dx\)
9. \(\int x^2e^{2x}\ dx\)
10. \(\int 2x^2e^{3x}\ dx\)
11. \(\int x^3\cos(3x)\ dx\)
12. \(\int xe^{-x}\ dx\)
13. \(\int e^{3x}\sin(2x)\ dx\)
14. \(\int \ln(x)\ dx\)
15. \(\int \log_{10}(x)/x^3\ dx\)

{Solutions: 1. \(x\sin x+\cos x + C\); 2. \(\frac{x^2}{2}\ln(x) - \frac{x^2}{4}+C\); 3. \(-\frac{x}{4}\cos(4x)+\frac{1}{16}\sin(4x)+C\); 4. \(\frac{x}{11}(1+x)^{11} - \frac{1}{132}(1+x)^{12}+C\); 5. \(\frac{x^8}{8}\ln(x) - \frac{x^8}{64}+C\); 6. \(\frac{2}{3}x^{3/2}\ln(x) - \frac{4}{9}x^{3/2}+C\); 7. \(x^2e^x - 2e^x(x-1)+C\); 8. \(x^2\sin(x) - 2(\sin(x)-x\cos(x))+C\); 9. \(\frac{x^2}{2}e^{2x}-\frac{x}{2}e^{2x}+\frac{1}{4}e^{2x}+C\); 10. \(\frac{2}{3}x^2e^{3x}-\frac{4}{9}xe^{3x}+\frac{4}{27}e^{3x}+C\); 11. \(\frac{1}{3}x^3\sin(3x)+\frac{1}{3}x^2\cos(3x)-\frac{2}{9}x\sin(3x)-\frac{2}{27}\cos(3x)+C\); 12. \(-xe^{-x}-e^{-x}+C\); 13. \(\frac{9}{13}\left( \frac{1}{3}e^{3x}\sin(2x)-\frac{2}{9}e^{3x}\cos(2x) \right)+C\); 14. \(x\ln(x) - x +C\); 15. \(-\frac{1}{2\ln(10)}x^{-2}\ln(x) - \frac{1}{4\ln(10)}x^{-2} + C\); }