Qualifying round 2014 answers PDF

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MIT Integration Bee Qualifying Exam 1

Z e 1

log( x2 ) dx = 2

√ sin( 3 x ) dx = 0

2

Z 9

3

Z ∞ d h 1+ x − x 2 i e dx = − e

4

5

6

−9

Z 2

r

Z √

Z

10

Z 0

11

Z

−1

x2 1 dx = − log 2 x−1 2

x arctan x dx =

x 1 2 x arctan( x ) − + 2 2

1 arctan( x ) 2

dx

0

0

21 January 2014

x+

xe

√

x

q

x+

√

dx = 2 e

x + · · · dx =

√

x

   x − 53  dx 1  log  = x + 38  91 x2 − 15x − 2014

13

Z

h i e x log(1 + x2 ) − 2(1 + x ) arctan x dx =

19 6

√ ( x − 2 x + 2)

sin(2x ) cos (3x ) dx =

12

Z

h i e x log(1 + x2 ) − 2x arctan x

cos x cos 5x − 10 2

14

Z

p (arcsin x )2 dx = 2 1 − x2 arcsin x −

2x + x (arcsin x )2 7

Z 2π 0

√ 2π +4 3 |1 + 2 sin x | dx = 3

15

Z √ 2 x −1

x

arctan 8

Z

x (1 − x )2014 dx

=

x )2016

(1 − 2016

−

(1 − x )2015 2015 9

Z

arcsinh( x ) = x arcsinh( x ) −

16

Z

p

dx

=

p

x2 − 1 −

x2 − 1

x sec2 (4x ) dx

=

x tan (4x ) + 4

log(cos (4x )) 16 p

x2 + 1 17

Z

2 dx = 2 log (2 − 6 − 11x + 6x2 − x3

x ) − log( x2 − 4x + 3)

18

Z 1 0

1

⌊1 − log2 (1 − x )⌋

dx = log 2

19

Z 1/ √3

20

Z 5π/2

0

0

q

x+

p

x2 + 1 dx =

√ dx 7 3π = 9 2 + cos x

2 3...