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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```