Mitintegrationbee2006.docx

MATH 1920-01

SPRING

CALCULUS II

2006

MIT Integration Bee The MIT Integration Bee is a yearly competition open to undergraduates. The initial phase of the competition consists of an hour-long written competition consisting of forty integration problems. The subsequent phases are handled like a Spelling Bee. At the end of the 1991 competition, there were four finalists, who were given the following four problems.

r ex(x 2) − x3

r

x sec2 x dx

r dx r

cot x dx log(sin x)

x sec x(x tan x + 2) dx.

The problems for the first round are given below. Have fun! r 1. e1991x dx r log x dx

2. r

(sin x − cos x)2 dx

3. r

2

esin xecos

4.

2

x

dx

r dx πx r x e +x e dx 6. 5.

r 7. 8.

r r

9.

x x4 + 1

dx

dx x log x e5x + e7x ex + e−x

dx

1

r 10.

r

11. r

xex sin x dx x+1 √ dx 3 2 x + 2x + 2 (cosh x − sinh x) dx

12. r 13. r

x2 + 1 x4 − x2 + 1

dx

sec3 x dx

14. 15. r

dx sec x + tan x sin x r '\ 16. −1 + 2/(1 + 3x) dx r 17. r 18.

sin x esec x dx cos2 x dx πx2 + tan−1 x + x2 tan−1 x + π

dx 19. r 12 + 13 cos x r 3 x +1 dx 20. x +1 x +1 r 21. e1991 dx 22. r

dx x2 − 10x + 26 23. r dx x2 − 11x − 26 r (1 − 4x4)−1/2 dx 24. (4x)−1 (4x)−1 r 25. (log x + 1)xx dx

2

r 26.

r

(cos 2x)(sin 6x) dx dx

√

27.

√ dx x(1 + x)

r

e1/xx−3 dx

28. r 29. r 30.

√

csc x − sin x dx

x2 + 1 dx x3 + 1

r 42x dx

31. r

x5ex dx

32. r

2

xex dx

33. 34. r 35. r r

dx (x2 + 1)2 dx ex + e−x tan x log | sec x| dx

36. r 37.

cos(sin x) cos x dx

38. r

dx x2 − 9

r 39.

√

π dx

16 − e2 r √ 40. tan x dx

3