Resolução de exercícios de integrais
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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