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# Resolução de exercícios de integrais

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ln|1 + \ud835\udc652| 2 \u222b 1 1 + \ud835\udc652 \ud835\udc51\ud835\udc65 = arctg(\ud835\udc65) \u222b \ud835\udc65 \u2212 1 1 + \ud835\udc652 \u22c5 cos(2\ud835\udf0b) \ud835\udc51\ud835\udc65 = ln|1 + \ud835\udc652| 2 \u2212 arctg(\ud835\udc65) + \ud835\udc36 ln|1 + \ud835\udc652| 2 \u2212 arctg(\ud835\udc65) + \ud835\udc36 g) \u222b cos3(\ud835\udc65)\ud835\udc51\ud835\udc65 \u222b cos2(\ud835\udc65) \u22c5 cos(\ud835\udc65) \ud835\udc51\ud835\udc65 \ud835\udc62 = cos2(\ud835\udc65) \ud835\udc51\ud835\udc62 = \u22122 cos(\ud835\udc65) \u22c5 sen(\ud835\udc65) \ud835\udc51\ud835\udc65 \ud835\udc51\ud835\udc63 = cos(\ud835\udc65) \ud835\udc51\ud835\udc65 \ud835\udc63 = sen(\ud835\udc65) \u222b \ud835\udc62 \ud835\udc51\ud835\udc63 = \ud835\udc62 \u22c5 \ud835\udc63 \u2212 \u222b \ud835\udc63 \ud835\udc51\ud835\udc62 \u222b cos2(\ud835\udc65) \u22c5 cos(\ud835\udc65) \ud835\udc51\ud835\udc65 = cos2(\ud835\udc65) \u22c5 sen(\ud835\udc65) \u2212 \u222b \u22122 cos(\ud835\udc65) \u22c5 sen(\ud835\udc65) \u22c5 sen(\ud835\udc65) \ud835\udc51\ud835\udc65 \u222b cos2(\ud835\udc65) \u22c5 cos(\ud835\udc65) \ud835\udc51\ud835\udc65 = cos2(\ud835\udc65) \u22c5 sen(\ud835\udc65) + \u222b 2 cos(\ud835\udc65) \u22c5 sen2(\ud835\udc65) \ud835\udc51\ud835\udc65 \u222b cos2(\ud835\udc65) \u22c5 cos(\ud835\udc65) \ud835\udc51\ud835\udc65 = cos2(\ud835\udc65) \u22c5 sen(\ud835\udc65) + 2 \u222b cos(\ud835\udc65) \u22c5 sen2(\ud835\udc65) \ud835\udc51\ud835\udc65 \u222b cos(\ud835\udc65) \u22c5 sen2(\ud835\udc65) \ud835\udc51\ud835\udc65 \ud835\udc64 = sen(\ud835\udc65) \ud835\udc51\ud835\udc64 = cos(\ud835\udc65) \ud835\udc51\ud835\udc65 \ud835\udc51\ud835\udc65 = \ud835\udc51\ud835\udc64 cos(\ud835\udc65) \u222b cos(\ud835\udc65) \u22c5 \ud835\udc642 \ud835\udc51\ud835\udc64 cos(\ud835\udc65) \u222b \ud835\udc642 \ud835\udc51\ud835\udc64 \ud835\udc643 3 sen3(\ud835\udc65) 3 \u222b cos2(\ud835\udc65) \u22c5 cos(\ud835\udc65) \ud835\udc51\ud835\udc65 = cos2(\ud835\udc65) \u22c5 sen(\ud835\udc65) + 2 \u22c5 sen3(\ud835\udc65) 3 + \ud835\udc36 cos2(\ud835\udc65) \u22c5 sen(\ud835\udc65) + 2 \u22c5 sen3(\ud835\udc65) 3 + \ud835\udc36 cos2(\ud835\udc65) \u22c5 sen(\ud835\udc65) + 2 sen3(\ud835\udc65) 3 + \ud835\udc36 3 cos2(\ud835\udc65) \u22c5 sen(\ud835\udc65) + 2 sen3(\ud835\udc65) 3 + \ud835\udc36 2 sen3(\ud835\udc65) + 3 sen(\ud835\udc65) cos2(\ud835\udc65) 3 + C h) \u222b \ud835\udc65 1+\ud835\udc654 \ud835\udc51\ud835\udc65 \u222b \ud835\udc65 1 + (\ud835\udc652)2 \ud835\udc51\ud835\udc65 \ud835\udc62 = \ud835\udc652 \ud835\udc51\ud835\udc62 = 2\ud835\udc65 \ud835\udc51\ud835\udc65 \ud835\udc51\ud835\udc65 = \ud835\udc51\ud835\udc62 2\ud835\udc65 \u222b \ud835\udc65 1 + \ud835\udc622 \ud835\udc51\ud835\udc62 2\ud835\udc65 \u222b 1 1 + \ud835\udc622 \ud835\udc51\ud835\udc62 2 1 2 \u222b 1 1 + \ud835\udc622 \ud835\udc51\ud835\udc62 1 2 \u22c5 arctg(\ud835\udc62) + \ud835\udc36 1 2 \u22c5 arctg(\ud835\udc652) + \ud835\udc36 arctg(\ud835\udc652) 2 + \ud835\udc36 i) \u222b ln(\ud835\udc65 + 3) \ud835\udc51\ud835\udc65 \ud835\udc62 = \ud835\udc65 + 3 \ud835\udc51\ud835\udc62 = \ud835\udc51\ud835\udc65 \u222b ln(\ud835\udc62) \ud835\udc51\ud835\udc62 \u222b ln(\ud835\udc62) = \ud835\udc62 \u22c5 ln(\ud835\udc62) \u2212 \ud835\udc62 (\ud835\udc65 + 3) \u22c5 ln(\ud835\udc65 + 3) \u2212 (\ud835\udc65 + 3) (\ud835\udc65 + 3) \u22c5 ln(\ud835\udc65 + 3) \u2212 \ud835\udc65 \u2212 3 + \ud835\udc36 \u22123 + \ud835\udc36 = \ud835\udc36 (\ud835\udc65 + 3) \u22c5 ln(\ud835\udc65 + 3) \u2212 \ud835\udc65 + \ud835\udc36 j) \u222b \ud835\udc522\ud835\udc65 \u22c5 cos(3\ud835\udc65) \ud835\udc51\ud835\udc65 \ud835\udc62 = \ud835\udc522\ud835\udc65 \ud835\udc51\ud835\udc62 = 2\ud835\udc522\ud835\udc65 \ud835\udc51\ud835\udc63 = cos(3\ud835\udc65) \ud835\udc51\ud835\udc65 \ud835\udc63 = \u222b \ud835\udc51\ud835\udc63 \u222b cos(3\ud835\udc65) \ud835\udc51\ud835\udc65 \ud835\udc64 = 3\ud835\udc65 \ud835\udc51\ud835\udc64 = 3 \ud835\udc51\ud835\udc65 \ud835\udc51\ud835\udc65 = \ud835\udc51\ud835\udc64 3 \ud835\udc63 = \u222b cos(\ud835\udc64) \ud835\udc51\ud835\udc64 3 \ud835\udc63 = sen(\ud835\udc64) 3 \ud835\udc63 = sen(3\ud835\udc65) 3 \u222b \ud835\udc62 \ud835\udc51\ud835\udc63 = \ud835\udc62 \u22c5 \ud835\udc63 \u2212 \u222b \ud835\udc63 \ud835\udc51\ud835\udc62 \u222b \ud835\udc522\ud835\udc65 \u22c5 cos(3\ud835\udc65) \ud835\udc51\ud835\udc65 = \ud835\udc522\ud835\udc65 \u22c5 sen(3\ud835\udc65) 3 \u2212 \u222b sen(3\ud835\udc65) \u22c5 2\ud835\udc522\ud835\udc65 3 \ud835\udc51\ud835\udc65 \u2212 2 3 \u222b sen(3\ud835\udc65) \u22c5 \ud835\udc522\ud835\udc65\ud835\udc51\ud835\udc65 \ud835\udc62 = \ud835\udc522\ud835\udc65 \ud835\udc51\ud835\udc62 = 2\ud835\udc522\ud835\udc65 \ud835\udc51\ud835\udc63 = sen(3\ud835\udc65) \ud835\udc51\ud835\udc65 \ud835\udc63 = \u222b \ud835\udc51\ud835\udc63 \ud835\udc63 = \u222b sen(3\ud835\udc65) \ud835\udc51\ud835\udc65 \u210e = 3\ud835\udc65 \ud835\udc51\u210e = 3 \ud835\udc51\ud835\udc65 \ud835\udc51\ud835\udc65 = \ud835\udc51\u210e 3 \ud835\udc63 = \u222b sen(\u210e) \ud835\udc51\u210e 3 \ud835\udc63 = \u2212 cos(\u210e) 3 \ud835\udc63 = \u2212 cos(3\ud835\udc65) 3 \u2212 2 3 \u222b sen(3\ud835\udc65) \u22c5 \ud835\udc522\ud835\udc65\ud835\udc51\ud835\udc65 = \u2212 2 3 ( \u2212\ud835\udc522\ud835\udc65 \u22c5 cos(3\ud835\udc65) 3 ) \u2212 2 3 \u22c5 (\u2212 \u222b \u22122\ud835\udc522\ud835\udc65 \u22c5 cos(3\ud835\udc65) 3 \ud835\udc51\ud835\udc65) 2\ud835\udc522\ud835\udc65 \u22c5 cos(3\ud835\udc65) 9 \u2212 \u222b 4\ud835\udc522\ud835\udc65 \u22c5 cos(3\ud835\udc65) 9 \ud835\udc51\ud835\udc65 \u222b \ud835\udc522\ud835\udc65 \u22c5 cos(3\ud835\udc65) \ud835\udc51\ud835\udc65 = \ud835\udc522\ud835\udc65 \u22c5 sen(3\ud835\udc65) 3 \u2212 \u222b sen(3\ud835\udc65) \u22c5 2\ud835\udc522\ud835\udc65 3 \ud835\udc51\ud835\udc65 \u222b \ud835\udc522\ud835\udc65 \u22c5 cos(3\ud835\udc65) \ud835\udc51\ud835\udc65 = \ud835\udc522\ud835\udc65 \u22c5 sen(3\ud835\udc65) 3 + 2\ud835\udc522\ud835\udc65 \u22c5 cos(3\ud835\udc65) 9 \u2212 \u222b 4\ud835\udc522\ud835\udc65 \u22c5 cos(3\ud835\udc65) 9 \ud835\udc51\ud835\udc65 \u222b 4\ud835\udc522\ud835\udc65 \u22c5 cos(3\ud835\udc65) 9 \ud835\udc51\ud835\udc65 + \u222b \ud835\udc522\ud835\udc65 \u22c5 cos(3\ud835\udc65) \ud835\udc51\ud835\udc65 = \ud835\udc522\ud835\udc65 \u22c5 sen(3\ud835\udc65) 3 + 2\ud835\udc522\ud835\udc65 \u22c5 cos(3\ud835\udc65) 9 4 9 \u222b \ud835\udc522\ud835\udc65 \u22c5 cos(3\ud835\udc65) \ud835\udc51\ud835\udc65 + \u222b \ud835\udc522\ud835\udc65 \u22c5 cos(3\ud835\udc65) \ud835\udc51\ud835\udc65 = \ud835\udc522\ud835\udc65 \u22c5 sen(3\ud835\udc65) 3 + 2\ud835\udc522\ud835\udc65 \u22c5 cos(3\ud835\udc65) 9 4 9 + 1 = 13 9 \ud835\udc522\ud835\udc65 13 \u22c5 [3 sen(3\ud835\udc65) + 2 cos(3\ud835\udc65)] + \ud835\udc36 13 9 \u222b \ud835\udc522\ud835\udc65 \u22c5 cos(3\ud835\udc65) \ud835\udc51\ud835\udc65 = \ud835\udc522\ud835\udc65 \u22c5 sen(3\ud835\udc65) 3 + 2\ud835\udc522\ud835\udc65 \u22c5 cos(3\ud835\udc65) 9 \u222b \ud835\udc522\ud835\udc65 \u22c5 cos(3\ud835\udc65) \ud835\udc51\ud835\udc65 = 9 13 \u22c5 ( \ud835\udc522\ud835\udc65 \u22c5 sen(3\ud835\udc65) 3 + 2\ud835\udc522\ud835\udc65 \u22c5 cos(3\ud835\udc65) 9 ) \u222b \ud835\udc522\ud835\udc65 \u22c5 cos(3\ud835\udc65) \ud835\udc51\ud835\udc65 = 9 13 \u22c5 \ud835\udc522\ud835\udc65 \u22c5 sen(3\ud835\udc65) 3 + 9 13 \u22c5 2\ud835\udc522\ud835\udc65 \u22c5 cos(3\ud835\udc65) 9 \u222b \ud835\udc522\ud835\udc65 \u22c5 cos(3\ud835\udc65) \ud835\udc51\ud835\udc65 = 3\ud835\udc522\ud835\udc65 \u22c5 sen(3\ud835\udc65) 13 + 2\ud835\udc522\ud835\udc65 \u22c5 cos(3\ud835\udc65) 13 \u222b \ud835\udc522\ud835\udc65 \u22c5 cos(3\ud835\udc65) \ud835\udc51\ud835\udc65 = \ud835\udc522\ud835\udc65 13 \u22c5 [3 sen(3\ud835\udc65) + 2 cos(3\ud835\udc65)] + \ud835\udc36 k) \u222b \ud835\udc52ln(\ud835\udc65 2) \u22c5 2\ud835\udc65 \ud835\udc51\ud835\udc65 \ud835\udc52ln(\ud835\udc4e) = \ud835\udc4e \u222b \ud835\udc652 \u22c5 2\ud835\udc65 \ud835\udc51\ud835\udc65 \u222b 2\ud835\udc653\ud835\udc51\ud835\udc65 = 2\ud835\udc654 4 \u222b 2\ud835\udc653\ud835\udc51\ud835\udc65 = \ud835\udc654 2 \ud835\udc654 4 + \ud835\udc36

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