840
Answer Key
319 . f ( t ) =2cos(3 t )−cos(2 t ); ⌠ ⌡ 0 π /2 ⎛
⎝ 2cos(3 t )−cos(2 t ) ⎞
⎠ = − 2 3
321 . −1 3
e −3 x + C
− x ln3 ⎛ ⎝ x 2
323 . − 3
+ C
⎞ ⎠ + C
325 . ln
327 . 2 x + C 329 . − 1 ln x
+ C
331 . ln ⎛ ⎠ + C 333 . ln( x cos x )+ C 335 . − 1 2 ⎛ ⎝ ln(cos( x )) ⎞ ⎝ ln(ln x ) ⎞
⎠ 2 + C
− x 3 3 +
337 . − e
C
339 . e tan x + C 341 . t + C 343 . 1 9 x 3 ⎛ ⎝ ln ⎛ ⎝ x 3 ⎞
⎞ ⎠ + C
⎠ −1
345 . 2 x (ln x −2)+ C 347 . ⌠ ⌡ 0 ln x e t dt = e t | 0 ln x
= e ln x − e 0 = x −1
349 . − 1 3 ln ⎛ 351 . − 1 2 ln | csc ⎛
⎝ sin(3 x )+cos(3 x ) ⎞ ⎠
⎞ ⎠ | + C
⎞ ⎠ +cot
⎛ ⎝ x 2
⎝ x 2
⎠ 2 + C
353 . − 1 2 ⎛ 355 . 1 357 . ln ⎛ 3 ln
⎝ ln(csc x ) ⎞
⎛ ⎝ 26 7
⎞ ⎠
⎝ 3−1 ⎞ ⎠
3 2
359 . 1
2 ln
361 . y −2ln | y +1 | + C 363 . ln | sin x −cos x | + C 365 . − 1 3 ⎛ ⎝ 1− ⎛ ⎝ ln x 2 ⎞ ⎠ ⎞ ⎠ 3/2 + C 367 . Exact solution: e −1
e , R 50 =0.6258. Since f is decreasing, the right endpoint estimate underestimates the area.
369 . Exact solution: 2ln(3) − ln(6) 2 ,
R 50 =0.2033. Since f is increasing, the right endpoint estimate overestimates the area.
371 . Exact solution: − 1 ln(4)
, R 50 =−0.7164. Since f is increasing, the right endpoint estimate overestimates the area (the
actual area is a larger negative number). 373 . 11 2 ln2 375 . 1 ln(65, 536) 377 . ⌠ ⌡ N N +1 xe − x 2 dx = 1 2 ⎛ ⎝ e − N 2
⎞ ⎠ . The quantity is less than 0.01 when N =2.
− e −( N +1) 2
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