1. EXERCÍCIOS M1 Anatolie Sochirca ACM DEETC ISEL
1
Cálculo de integrais indefinidos com aplicação das propriedades e das
fórmulas da tabela dos integrais (integração imediata).
Para aplicar o método de integração imediata transformamos a expressão sob
o sinal integral com o objectivo de obter um integral ou uma soma algébrica de
integrais da tabela dos integrais.
Neste caso é útil a transformação: se )(xG é uma primitiva evidente da função
)(xg e ))(()( xGuxf = então
( )∫∫∫ ⋅=⋅′⋅=⋅⋅ )())(()())(()()( xGdxGuxdxGxGuxdxgxf .
►1) =⋅








−−+
⋅
−=⋅
−−+⋅−
∫∫ xd
xx
x
x
x
x
xx
x
x
xd
x
xxxxx
33
5 3
333
3
3
5 33
223223
=⋅







−−+−=⋅










−−+
⋅
−= ∫∫
−−−−+−
xdxxxxxxd
xx
x
x
x
x
xx
x
x 3
1
3
1
5
3
3
1
1
3
1
2
1
1
3
1
3
3
1
3
1
5
3
3
1
3
1
2
1
3
1
3
223
223
=⋅







−−+−= ∫
−
xdxxxxx 3
1
15
4
3
2
6
7
3
8
223
∫∫∫∫∫ =⋅−⋅−⋅+⋅−⋅=
−
xdxxdxxdxxdxxdx 3
1
15
4
3
2
6
7
3
8
223
∫∫∫∫∫ =⋅⋅−⋅−⋅⋅+⋅⋅−⋅=
−
xdxxdxxdxxdxxdx 3
1
15
4
3
2
6
7
3
8
223
=+
+−
⋅−
+
−
+
⋅+
+
⋅−
+
=
+−++++
C
xxxxx
1
3
1
2
1
15
4
1
3
2
2
1
6
7
3
1
3
8
1
3
1
1
15
4
1
3
2
1
6
7
1
3
8
=+⋅−−⋅+⋅−= C
xxxxx
3
2
2
15
19
3
5
2
6
13
3
3
11
3
2
15
19
3
5
6
13
3
11
=+⋅⋅−⋅−⋅⋅+⋅⋅−⋅= Cxxxxx 3
2
15
19
3
5
6
13
3
11
2
3
2
19
15
5
3
2
13
6
3
11
3
Cxxxxx +⋅−⋅−⋅+⋅−⋅= 3
2
15
19
3
5
6
13
3
11
3
19
15
5
6
13
18
11
3
. ■

2. EXERCÍCIOS M1 Anatolie Sochirca ACM DEETC ISEL
2
►2) =⋅
+
=⋅=⋅=⋅
⋅ ∫∫∫∫ )(
5
1
)(
)5(
1
)5(
1
)5(
1
xnld
xnlnl
xnld
xnlx
xd
xnl
xd
xnlx
( ) ( ) CxnlnlCxnlnlnlxnlnld
xnlnl
+=++=+⋅
+
= ∫ )5(5)5(
5
1
. ■
Outro método:
► =
⋅
⋅
⋅=
⋅
⋅
⋅=⋅
⋅⋅
=⋅
⋅ ∫∫∫∫ x
xd
xnlx
xd
xnl
xd
xnlx
xd
xnlx 5
)5(
)5(
1
5
5
)5(
1
)5(5
5
)5(
1
( ) Cxnlnlxnld
xnl
+=⋅= ∫ )5()5((
)5(
1
. ■
►3) ( )=⋅=
+
⋅=⋅
+ ∫∫∫ xarctgdxarctg
x
xd
xarctgxd
x
xarctg 3
2
3
2
3
11
( ) CxarctgC
xarctg
+⋅=+
+
=
+
4
13
4
1
13
. ■
►4) =
+++
⋅+⋅=⋅
++
+⋅
∫∫ 196
)3(2
106
)3(2
22
xx
xd
xarctgxd
xx
xarctg
( )=+⋅+⋅=
++
+
⋅+⋅= ∫∫ )3()3(2
1)3(
)3(
)3(2 2
xarctgdxarctg
x
xd
xarctg
CxarctgC
xarctg
++=+
+
+
⋅=
+
)3(
11
)3(
2 2
11
. ■
►5)
( )
( )
( ) ( )
=
+
⋅−⋅
=
+
⋅−
=⋅
+
−
∫∫∫ 222222
)2()2()2(
xsenxosc
xdxoscxdxsen
xsenxosc
xdxoscxsen
xd
xsenxosc
xoscxsen
( ) ( )
=
+
−−⋅⋅
=
+
⋅−⋅⋅⋅
= ∫∫ 2222
)()(22
xsenxosc
senxdxoscdxosc
xsenxosc
xdxoscxdxoscxsen
( ) ( )
=
+
+⋅⋅
−=
+
−⋅⋅−
= ∫∫ 2222
)()(2)()(2
xsenxosc
senxdxoscdxosc
xsenxosc
senxdxoscdxosc
( ) ( )
=
+
+
−=
+
+
−= ∫∫ 22
2
22
2
)()()(
xsenxosc
senxxoscd
xsenxosc
senxdxoscd

3. EXERCÍCIOS M1 Anatolie Sochirca ACM DEETC ISEL
3
( ) ( ) =+
+−
+
−=+⋅+−=
+−
−
∫ C
xsenxosc
senxxoscdxsenxosc
12
)(
122
222
( ) C
xsenxosc
C
xsenxosc
+
+
=+
−
+
−=
−
2
12
1
1
. ■
►6)
( ) =
−
⋅
′
⋅=
−
⋅⋅
⋅=
−
⋅⋅⋅
=⋅
−
∫∫∫∫ 2
2
222
12
1
1
2
2
1
1
2
2
1
1 x
xdx
x
xdx
x
xdx
xd
x
x
( )
=
−
−
⋅−=
−
−
⋅−=
−
−−
⋅=
−
⋅= ∫∫∫∫
2
1
2
2
2
2
2
2
2
2
1
)1(
2
1
1
)1(
2
1
1
)(
2
1
1
)(
2
1
x
xd
x
xd
x
xd
x
xd
( ) ( ) =+
+−
−
⋅−=−⋅−⋅−=
+−
−
∫ C
x
xdx
1
2
1
1
2
1
)1(1
2
1
1
2
1
2
22
1
2
( ) CxCx +−−=+−−= 22
1
2
11 . ■
►7)
( ) =
−
⋅
′
⋅=
−
⋅⋅
⋅=
−
⋅⋅⋅
=⋅
−
∫∫∫∫ 4
2
444
12
1
1
2
2
1
1
2
2
1
1 x
xdx
x
xdx
x
xdx
xd
x
x
( ) ( )
Cxarcsen
x
xd
x
xd
+⋅=
−
⋅=
−
⋅= ∫∫ )(
2
1
1
)(
2
1
1
)(
2
1 2
22
2
22
2
. ■
►8) =⋅







−
+
−
⋅
=⋅
−
+⋅
∫∫ xd
x
x
x
xarcsen
xd
x
xxarcsen
222
11
2
1
2
=⋅
−
+⋅
−
⋅⋅=⋅
−
+⋅
−
⋅
= ∫∫∫∫
44 344 21
)6
2222
11
1
2
11
2
exemplover
xd
x
x
xd
x
xarcsenxd
x
x
xd
x
xarcsen
( ) =+−−+⋅
′
⋅⋅= ∫ Cxxdxarcsenxarcsen 2
12
CxxarcsenCxxarcsen +−−=+−−⋅⋅= 2222
11
2
1
2 . ■

4. EXERCÍCIOS M1 Anatolie Sochirca ACM DEETC ISEL
4
►9)
( ) =
+⋅⋅
⋅+⋅⋅
=
+⋅⋅
⋅+⋅
=⋅
+⋅⋅
+⋅
∫∫∫ xxx
xdxdx
xxx
xdx
xd
xxx
x
2
13
2
13
2
13
=
+⋅
+







⋅
=
+⋅
+⋅
′








⋅⋅
=
+⋅⋅
+⋅⋅
= ∫∫∫
xx
xdxd
xx
xdxdx
xxx
xdxdx
2
3
2
3
2
3
2
3
2
1
2
1
2
2
2
3
2
3
2
3
Cxxnl
xx
xxd
xx
xdxd
++⋅=
+⋅








+⋅
=
+⋅
+







⋅
= ∫∫ 2
3
2
3
2
3
2
3
2
3
2
2
2
2
2
. ■
►10) ( ) ( ) ( ) ( ) ( )=⋅+=⋅
′
⋅+=⋅⋅+ ∫∫∫
xxxxxx
edexdeexdee
πππ
222
( ) ( ) ( ) C
e
ede
x
xx
+
+
+
=+⋅+=
+
∫ 1
2
22
1
π
π
π
. ■
►11) ( ) ( )( ) ( ) =⋅⋅
⋅
=⋅
′
⋅⋅
⋅
=⋅⋅=⋅⋅ ∫∫∫∫
xxxxx
ed
enl
xde
enl
xdexde 3
)3(
1
3
)3(
1
33
( ) ( ) ( ) C
nl
e
C
enlnl
e
Ce
enl
xx
x
+
+
⋅
=+
+
⋅
=+⋅⋅
⋅
=
13
3
3
3
3
)3(
1
. ■
►12) =⋅





−=⋅
−
=⋅=⋅ ∫∫∫∫ xd
xosc
xosc
xosc
xd
xosc
xosc
xd
xosc
xsen
xdxtg 2
2
22
2
2
2
2 11
( ) =+−⋅
′
=⋅−⋅=⋅





−= ∫∫∫∫ Cxxdxtgxdxd
xosc
xd
xosc
1
1
1
1
22
( ) CxxtgCxxtgd +−=+−= ∫ . ■
►13) Ceed
x
dexd
x
exd
x
e xxxx
x
+−=







−=





⋅−=⋅
′






⋅−=⋅ ∫∫∫∫
1111
2
1
11
. ■

5. EXERCÍCIOS M1 Anatolie Sochirca ACM DEETC ISEL
5
►14) ( ) ( )=⋅⋅=⋅
′
⋅⋅=⋅⋅⋅⋅=⋅⋅ ∫∫∫∫
222222
7
2
1
2
1
72
2
1
77 xdxdxxdxxdx xxxx
( ) ( ) C
nl
d
nl
xdnl
nl
x
xx
+
⋅
=⋅⋅=⋅⋅⋅⋅= ∫∫ 72
7
7
7
1
2
1
77
7
1
2
1
2
222
. ■
►15) =




 −
=⇔⋅−==⋅∫ 2
)2(1
21)2()6( 222 α
ααα
osc
sensenoscxdxsen
=⋅−⋅=⋅





−=⋅
−
= ∫∫∫∫ xd
xosc
xdxd
xosc
xd
xosc
2
)12(
2
1
2
)12(
2
1
2
)12(1
( ) =⋅
′
⋅⋅−⋅=⋅⋅−⋅= ∫∫∫∫ xdxsenxdxdxoscxd )12(
12
1
2
1
2
1
)12(
2
1
2
1
( ) C
xsenx
xsendxd +−=⋅−⋅= ∫∫ 24
)12(
2
)12(
24
1
2
1
. ■
►16) =⋅
⋅
+
=⋅
⋅
=
⋅ ∫∫∫ xd
xoscxsen
xoscxsen
xd
xoscxsenxoscxsen
xd 22
1
=⋅





+=⋅





⋅
+
⋅
= ∫∫ xd
xsen
xosc
xosc
xsen
xd
xoscxsen
xosc
xoscxsen
xsen 22
( ) ( ) =⋅
′
+⋅
′
−=⋅+⋅= ∫∫∫∫ xd
xsen
xsen
xd
xosc
xosc
xd
xsen
xosc
xd
xosc
xsen
( ) ( ) ( ) ( ) =⋅
′
−⋅
′
=⋅
′
−⋅
′
= ∫∫∫∫ xdxoscnlxdxsennlxd
xosc
xosc
xd
xsen
xsen
CxtgnlC
xosc
xsen
nlCxoscnlxsennl +=+=+−= . ■
►17)
( ) ( ) =
+
⋅
′
+
−=
+
⋅
′
−=⋅
+ ∫∫∫ xosc
xdxosc
xosc
xdxosc
xd
xosc
senx
5
5
55
( ) Cxoscnlxdxoscnl ++−=⋅
′
+−= ∫ 55 . ■

6. EXERCÍCIOS M1 Anatolie Sochirca ACM DEETC ISEL
6
►18) =⋅⋅=⋅
−
⋅⋅⋅
=⋅
−
⋅
∫∫∫ xd
xosc
xsen
xd
xsenxosc
xoscsenx
xd
xsenxosc
xoscsenx
)2(
)2(
2
1
2
2
1
2222
( )
( ) ( ) =⋅⋅−=⋅
′
⋅−
⋅= ∫∫
−
)2()2(
4
1
)2(
)2(
2
1
2
1
2
1
xoscdxoscxd
xosc
xosc
( ) ( ) ( ) CxoscC
xosc
C
xosc
+⋅−=+⋅−=+
+−
⋅−=
+−
2
12
1
1
2
1
)2(
2
1
2
1
)2(
4
1
1
2
1
)2(
4
1
. ■
Outro método:
►
( ) ( )
=
⋅−
⋅−⋅−
⋅=
−
⋅
′
⋅⋅⋅
=⋅
−
⋅
∫∫∫ xsen
xsend
xsenxosc
xdsenxsenx
xd
xsenxosc
xoscsenx
2
2
2222
21
2
2
1
2
1
2
2
1
( ) ( ) ( )=⋅−⋅⋅−⋅−=
⋅−
⋅−
⋅−= ∫∫
−
xsendxsen
xsen
xsend 22
1
2
2
2
2121
4
1
21
21
4
1
( ) ( ) ( ) CxoscCxsenC
xsen
+⋅−=+⋅−⋅=+
+−
⋅−
⋅−=
+−
2
1
2
1
2
1
2
1
2
)2(
2
1
21
2
1
1
2
1
21
4
1
. ■
►19) =⋅⋅⋅⋅⋅+=⋅⋅⋅+ ∫∫ xdxoscxsenxoscxdxsenxosc 254)2(54 22
( ) ( ) =⋅
′
⋅⋅⋅⋅+−=⋅
′
−⋅⋅⋅⋅+= ∫∫ xdxoscxoscxoscxdxoscxoscxosc 254254 22
( ) ( )=⋅⋅⋅+⋅−=⋅⋅+−= ∫∫ xoscdxoscxoscdxosc 2222
554
5
1
54
( ) ( ) ( ) =+
+
⋅+
⋅−=⋅+⋅⋅+⋅−=
+
∫ C
xosc
xoscdxosc
1
2
1
54
5
1
5454
5
1
1
2
1
2
22
1
2
( ) Cxosc +⋅+⋅−= 2
3
2
54
15
2
. ■