Uno por uno

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2009 Integration Bee Qualifier Solutions
1. xee +x dx x2 Solution: 1 ee 2 x−1/2 log x dx Solution: 2x1/2 log x −4x1/2
√ dx 4x2 −2
x2 2

2.

3.

Solution: 4.
π π/2

1 2

log(2x +

√ 4x2 − 2) dx

sin x log x −Solution: log π sin(log x)dx 1 Solution: 1 + 2 eπ 2
eπ 1

cos x x

5.

6.

sin2 (π(x − x )) dx Solution:666
3 x4 +4x3 +6x2 +4x+1 dx 2 x3 −3x2 +3x−1 63 Solution: 2 + 24 log 2 π/2 0 (sin x cos x

1332 0

7.

8.+ 2 cos2 x − 1) dx

Solution: 1/2 9. dx Solution: 2 − π
∞ e−x/2 √ 0 2 πx 2 x2 −4 0 x2 +4

10.

dx
1 2Solution: 11.
e 2 1 (log x)

dx Solution: e − 2 dx Solution: tanh x
dx 1+x+x2 +x3 Solution: 1 2 π/10 0 1 cosh2x

12.

13.

arctan x + 1 log(1 + x) − 1 log(1 + x2 ) 2 4

14.

sin2 (5x) dx Solution: π/20

15.sin x cos x dx Solution: 1 (1 + eπ/2 ) 8 dx Solution: π/4 dx Solution: − log(4 + x) + 2 log(5 + x) sin3 x cos2 x dxSolution: 2/15 x3 cos(x2 ) dx Solution: 1 cos(x2 ) + 1 x2 sin(x2 ) 2 2 dx Solution: cos(x1/6 ) + x1/6 sin(x1/6 )cos(x1/6 ) 6x2/3 π π/2 x+3 x2 +9x+20 π/2 cos x 0 sin x+cos x

π/4 2x e 0

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