Exercicios_Integracao_Indefinida

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Cálculo Diferencial e Integral I
Ficha de Exerćıcios n.o 5
Integração Indefinida
1. Calcule os seguintes integrais indefinidos imediatos ou quase imediatos:
1.1
∫
1
x3
dx
1.2
∫ (
9x2 +
1√
x3
)
dx
1.3
∫
(αx4 + βx3 + 3γ)dx
1.4
∫ (
1√
x
+
x
√
x
3
)
dx
1.5
∫
(2x2 − 3)2dx
1.6
∫
1
sen2 x
dx
1.7
∫ (√
2y − 1√
2y
)
dy
1.8
∫ √
2
3t2 + 3
dt
1.9
∫
x3
√
xdx
1.10
∫
x5 + 2x2 − 1
x4
dx
1.11
∫
x2
x2 + 1
dx
1.12
∫
x2 + 1
x2
dx
1.13
∫
senx
cos2 x
dx
1.14
∫ √
9
1− x2
dx
1.15
∫ √
4
x4 − x2
dx
1.16
∫
8x4 − 9x3 + 6x2 − 2x+ 1
x2
dx
1.17
∫ (
et
2
+
√
t+
1
t
)
dt
1.18
∫
cos θ tg θdθ
1.19
∫ (
ex − e−x
)
dx
1.20
∫
sec2 x
(
cos3 x+ 1
)
dx
1.21
∫
dx
a2x2 + a2
, a ∈ R \ {0}
1.22
∫
x2 − 1
x2 + 1
dx
1.23
∫
3
√
8(t− 2)6
(
t+
1
2
)3
dt
1.24
∫
lnx
x lnx2
dx
1.25
∫
tg2 x cosec2 xdx
2. Calcule os seguintes integrais indefinidos, usando a substituição indicada:
Página 2
2.1
∫
x sen(x2)dx (u = x2)
2.2
∫
1
(1 +
√
x)
√
x
dx (u =
√
x)
2.3
∫
cosx
1 + sen2 x
dx (u = senx)
2.4
∫
ex√
1− e2x
dx (u = ex)
2.5
∫
x
1 +
√
x
dx (u =
√
x)
2.6
∫
dx√
x+ 3
√
x
(x = u6)
2.7
∫ √
x√
−1 +
√
x
dx (u2 = −1 +
√
x)
2.8
∫
lnx
x
√
1 + ln x
dx (u2 = 1 + lnx)
2.9
∫
1 + tg2 x√
−1 + tg x
dx (u2 = −1 + tg x)
2.10
∫
x(x+ 3)10dx (u = x+ 3)
2.11
∫
1
x ln3 x
dx (u = lnx)
2.12
∫
sen(2x)
1 + sen3 x
dx (u = senx)
2.13
∫
arcsen (
√
x)√
x(1− x)
dx (u = arcsen (
√
x)
2.14
∫
ln(2x)
x ln(4x)
dx (u = lnx)
3. Usando uma substituição adequada, determine cada um dos seguintes integrais indefinidos:
3.1
∫
x√
1− x2
dx
3.2
∫
x2
3
√
x3 + 1
dx
3.3
∫
x
4 + x2
dx
3.4
∫
x cos(x2 + 1)dx
3.5
∫
x2 cos(x3 − 1)dx
3.6
∫
x3
x8 + 1
dx
3.7
∫
x√
1− x4
dx
3.8
∫
e
√
x
√
x
dx
3.9
∫
1
(1 + x)
√
x
dx
3.10
∫ sen(1
x
)
x2
dx
3.11
∫
1
x2
√
1− 1
x2
dx
3.12
∫
cos(cosx) senxdx
3.13
∫
cosx senxdx
3.14
∫
cosx
sen2 x
dx
3.15
∫
lnx
x
dx
3.16
∫
cos(lnx)
x
dx
3.17
∫
exee
x
dx
3.18
∫
ex+1
1 + ex
dx
3.19
∫
ex
1 + e2x
dx
3.20
∫
(arcsenx)3√
1− x2
dx
3.21
∫ √
arccosx√
1− x2
dx
4. Calcule os seguintes integrais indefinidos, usando o método de integração por partes.
Página 3
4.1
∫
x sen(5x)dx
4.2
∫
ln(1− x)dx
4.3
∫
te4tdt
4.4
∫
(x+ 1) cos(2x)dx
4.5
∫
x ln(3x)dx
4.6
∫
cos3 xdx
4.7
∫
ex cos
(x
2
)
dx
4.8
∫ √
x lnxdx
4.9
∫
cosec3 xdx
4.10
∫
x2 cos(ax)dx, com a 6= 0
4.11
∫
x cosec2 xdx
4.12
∫
arccotg(2x)dx
4.13
∫
eax sen(bx)dx, com a 6= 0 e b 6= 0
4.14
∫
ln(ax+ b)√
ax+ b
dx, com a 6= 0
4.15
∫
x3
√
1− x2dx
4.16
∫
ln3(2x)dx
4.17
∫
arctg(ax)dx, com a 6= 0
4.18
∫
x3 sen(4x)dx
4.19
∫
(x− 1)e−xdx
4.20
∫
x2 lnxdx
4.21
∫
x2exdx
4.22
∫
arcsen
(x
2
)
dx
4.23
∫
(x− 1) sec2 xdx
4.24
∫
e3x cos(4x)dx
4.25
∫
xn lnxdx, com n ∈ N
4.26
∫
ln(x2 + 1)dx
4.27
∫
ln(x+
√
x2 + 1)dx
4.28
∫
x arctg xdx
4.29
∫
x5ex
2
dx
4.30
∫
x cos2 xdx
4.31
∫
(x+ 3)2exdx
4.32
∫
x
√
x+ 1dx
4.33
∫
cos(lnx)dx
4.34
∫
arccosxdx
4.35
∫
sec3 xdx
4.36
∫
1
x3
e
1
xdx
5. Calcule os seguintes integrais indefinidos envolvendo funções trigonométricas:
5.1
∫
sen(2x) cos(4x)dx
5.2
∫
cos(3x) cos(2x)dx
5.3
∫
sen(5x) senxdx
5.4
∫
sen(3x) cos(5x)dx
Página 4
5.5
∫
sen(10x) sen(15x)dx
5.6
∫
cos
(x
2
)
cos
(x
3
)
dx
5.7
∫
sen
(x
3
)
cos
(
2x
3
)
dx
5.8
∫
cos(ax+ b) cos(ax− b)dx, a 6= 0
5.9
∫
sen(ωt) sen(ωt+ θ)dx, ω 6= 0
5.10
∫
cos3(2x)dx
5.11
∫
tg5(2x)dx
5.12
∫
cotg2
(x
5
)
dx
5.13
∫
sec6 xdx
5.14
∫
sen5 xdx
5.15
∫
cos2
(x
4
)
dx
5.16
∫
cos4 xdx
5.17
∫
cotg4 xdx
5.18
∫
cosec4 xdx
5.19
∫
tg3 xdx
5.20
∫
(sec2 x+ sen2(2x)) sec2 xdx
5.21
∫
cos3 x
sen4 x
dx
5.22
∫
sen2 x cos2 xdx
5.23
∫
sen3 x cos2 xdx
5.24
∫
sen4 x cos2 xdx
5.25
∫
tg5 x sec3 xdx
5.26
∫
sen2(3x) cos2(3x)dx
5.27
∫
sen2(2x) cos4(2x)dx
5.28
∫
tg4
(x
2
)
tg3
(x
2
)
dx
5.29
∫
cos2 x
sen6 x
dx
5.30
∫
cotg3 xdx
5.31
∫
1
sen5 x
dx
6. Determine os seguintes integrais indefinidos, usando uma substituição trigonométrica apropriada:
6.1
∫ √
9− x2dx
6.2
∫
dx√
4 + x2
6.3
∫
dx
x
√
4− x2
6.4
∫
dx
x
√
9 + x2
6.5
∫
dx
x2
√
x2 − 25
6.6
∫
xdx√
4− x2
6.7
∫
dx√
(x2 − 1)3
6.8
∫
dx
(36 + x2)2
6.9
∫
dx√
9− 4x2
6.10
∫
xdx
(16− x2)2
6.11
∫
x3dx√
9x2 + 49
6.12
∫
dx
x4
√
x2 − 3
Página 5
6.13
∫ √
1− 4x2dx
6.14
∫
x2dx√
1− x2
6.15
∫ √
9− (x− 1)2dx
6.16
∫
dx
x2
√
1 + x2
6.17
∫
dx√
x2 − 3x+ 2
6.18
∫
dx√
5 + 4x− x2
6.19
∫
dx
4x2 + 8x+ 13
6.20
∫ √
3− 2x− x2dx
6.21
∫ √
x2 − 2x+ 2dx
6.22
∫ √
x2 + xdx
6.23
∫
dx
2x2 − 4x+ 9
dx
6.24
∫
dx√
e2x + ex + 1
7. Calcule os seguintes integrais indefinidos de funções racionais:
7.1
∫
5x− 12
x(x− 4)
dx
7.2
∫
6x− 11
(x− 1)2
dx
7.3
∫
x+ 16
x2 + 2x− 8
dx
7.4
∫
2x2 − 25x− 33
(x+ 1)2(x− 5)
dx
7.5
∫
9x4 + 17x3 + 3x2 − 8x+ 3
x5 + 3x4
dx
7.6
∫
x3 + 6x2 + 3x+ 16
x3 + 4x
dx
7.7
∫
x3 + 3x− 2
x2 − x
dx
7.8
∫
x6 − x3 + 1
x4 + 9x2
dx
7.9
∫
4x3 + 2x2 − 5x− 18
(x− 4)(x+ 1)3
dx
7.10
∫
x3 + 3x2 + 3x+ 63
(x2 − 9)2
dx
7.11
∫
x
x2 − 5x+ 6
dx
7.12
∫
x2 + 3x+ 1
x2 − 2x− 3
dx
7.13
∫
x2 + 1
(x− 2)3
dx
7.14
∫
x4 + x+ 1
x3 − x
dx
7.15
∫
x+ 3
x3 − 2x2 − x+ 2
dx
7.16
∫
x− 1
x2(x+ 1)2
dx
7.17
∫
3
(x2 − 1)(x2 − 4)
dx
7.18
∫
4x2 + 17x+ 13
(x− 1)(x2 + 6x+ 10)
dx
7.19
∫
4x+ 1
x2 + 6x+ 8
dx
7.20
∫
x2
(x+ 2)2(x+ 4)2
dx
7.21
∫
1
x(x2 + 1)
dx
7.22
∫
2x2 − 3x− 3
(x− 1)(x2 − 2x+ 5)
dx
7.23
∫
x3 − 6
x4 + 6x2 + 8
dx
7.24
∫
3x− 7
x3 + x2 + 4x+ 4
dx
7.25
∫
4
x4 + 1
dx
7.26
∫
x5
x3 − 1
dx
7.27
∫
x3 + x− 1
(x2 + 2)2
dx
7.28
∫
4x2 − 8x
(x− 1)2(x2 + 1)2
dx
7.29
∫
1
(x2 − x)(x2 − x+ 1)2
dx
Página 6
Tabela de integrais imediatos
Seja u uma função de x e C ∈ R uma constante.
1.
∫
0dx = C
2.
∫
dx = x+ C
3.
∫
u′undx =
un+1
n+ 1
+ C, n ∈ R ∧ n 6= −1
4.
∫
u′audx =
au
ln a
+ C, a ∈ R+ \ {1}
5.
∫
u′eudx = eu + C
6.
∫
u′
u
dx = ln |u|+ C
7.
∫
u′ senudx = − cosu+ C
8.
∫
u′ cosudx = senu+ C
9.
∫
u′ sec2 udx = tg u+ C
10.
∫
u′ cosec2 udx = − cotg u+ C
11.
∫
u′ secu tg udx = secu+ C
12.
∫
u′ cosecu cotg udx = − cosecu+ C
13.
∫
u′
1 + u2
dx = arctg u+ C
14.
∫
u′√
1− u2
dx = arcsenu+ C
15.
∫
u′
u
√
u2 − 1
dx = arcsecu+ C
16.
∫
u′ secudx = ln | secu+ tg u|+ C
17.
∫
u′ cosecudx = − ln | cosecu+ cotg u|+ C
Página 7
Soluções
1.
1.1 − 1
2x2
+ C, C ∈ R
1.2 3x3 − 2√
x
+ C, C ∈ R
1.3
α
5
x5 +
β
4
x4 + 3γx+ C, C ∈ R
1.4 2
√
x+
2
15
√
x5 + C, C ∈ R
1.5
4
5
x5 − 4x3 + 9x+ C, C ∈ R
1.6 − cotg x+ C, C ∈ R
1.7
√
2y
(
2
3
y − 1
)
+ C, C ∈ R
1.8
√
2
3
arctg t+ C, C ∈ R
1.9
2
9
√
x9 + C, C ∈ R
1.10
x2
2
− 2
x
+
1
3x3
+ C, C ∈ R
1.11 x− arctg x+ C, C ∈ R
1.12 x− 1
x
+ C, C ∈ R
1.13 secx+ C, C ∈ R
1.14 3 arcsenx+ C, C ∈ R
1.15 2 arcsecx+ C, C ∈ R
1.16
8
3
x3 − 9
2
x2 + 6x− 2 ln |x| − 1
x
+ C, C ∈ R
1.17
1
2
et +
2
3
√
t3 + ln |t|+ C, C ∈ R
1.18 − cos θ + C, C ∈ R
1.19 ex + e−x + C, C ∈ R
1.20 senx+ tg x+ C, C ∈ R
1.21
1
a2
arctg x+ C, C ∈ R
1.22 x− 2 arctg x+ C, C ∈ R
1.23
1
2
t4 − 7
3
t3 + 2t2 + 4t+ C, C ∈ R
1.24
1
2
ln |x|+ C, C ∈ R
1.25 tg x+ C, C ∈ R
2.
2.1 −1
2
cos(x2) + C, C ∈ R
2.2 2 ln(1 +
√
x) + C, C ∈ R
2.3 arctg(sen x) + C, C ∈ R
2.4 arcsen(ex) + C, C ∈ R
2.5
2
√
x3
3
− x+ 2
√
x− 2 ln |1 +
√
x|+ C, C ∈ R
2.6 2
√
x− 3 3
√
x+ 6 6
√
x− 6 ln |1 + 6
√
x|+ C, C ∈ R
2.7
4
5
(√√
x− 1
)5
+
8
3
(√√
x− 1
)3
+ 4
√√
x− 1 + C, C ∈ R
2.8
2
3
(√
1 + ln x
)3 − 2√1 + ln x+ C, C ∈ R
2.9 2
√
tg x− 1 + C, C ∈ R
2.10
(x+ 3)12
12
− 3 (x+ 3)
11
11
+ C, C ∈ R
2.11 − 1
2 ln2 x
+ C, C ∈ R
2.12 ln(1 + sen2 x) + C, C ∈ R
2.13 arcsen2
√
x+ C, C ∈ R
2.14 lnx− ln 2 ln |ln(4x)|+ C, C ∈ R
3.
Página 8
3.1 −
√
1− x2 + C, C ∈ R
3.2
2
3
√
x3 + 1 + C, C ∈ R
3.3
1
2
ln(x2 + 4) + C, C ∈ R
3.4
1
2
sen(x2 + 1) + C, C ∈ R
3.5
1
3
sen(x3 − 1) + C, C ∈ R
3.6
arctg x4
4
+ C, C ∈ R
3.7
1
2
arcsen(x2) + C, C ∈ R
3.8 2e
√
x + C, C ∈ R
3.9 2 arctg
√
x+ C, C ∈ R
3.10 cos
(
1
x
)
+ C, C ∈ R
3.11 − arcsen
(
1
x
)
+ C, C ∈ R
3.12 − sen(cosx) + C, C ∈ R
3.13
1
2
sen2 x+ C,C ∈ R
3.14 − cosecx+ C, C ∈ R
3.15
1
2
ln2 x+ C, C ∈ R
3.16 sen(lnx) + C, C ∈ R
3.17 eex + C, C ∈ R
3.18 e ln(ex + 1) + C, C ∈ R
3.19 arctg(ex) + C, C ∈ R
3.20
(arctg x)4
4
+ C, C ∈ R
3.21 −2
3
√
(arccosx)3 + C, C ∈ R
4.
4.1 −x cos(5x)
5
+
sen(5x)
25
+ C, C ∈ R
4.2 (x− 1) ln(1− x)− x+ C, C ∈ R
4.3
e4t
4
(
t− 1
4
)
+ C, C ∈ R
4.4
(x+ 1) sen(2x)
2
+
cos(2x)
4
+ C, C ∈ R
4.5
x2
2
(
ln(3x)− 1
2
)
+ C, C ∈ R
4.6 cos2 x senx+
2 sen3 x
3
+ C, C ∈ R
4.7
2ex
5
(
sen
x
2
+ 2 cos
x
2
)
+ C, C ∈ R
4.8
2
3
x
√
x lnx− 4
9
x
√
x+ C, C ∈ R
4.9 −cosecx cotg x
2
+
x ln | cosecx− cotg x|
2
+ C, C ∈ R
4.10
(
x2
a
− 2
a3
)
sen(ax) +
2x
a2
cos(ax) + C, C ∈ R
4.11 −x cotg x+ ln | senx|+ C, C ∈ R
4.12 x arccotg(2x) +
ln(1 + 4x2)
4
+ C, C ∈ R
4.13
beax
a2 + b2
(a
b
sen(bx)− cos(bx)
)
+ C, C ∈ R
4.14
2
a
(ln(ax+ b)− 2)
√
ax+ b+ C, C ∈ R
4.15
(
−x
2
3
− 2
15
(1− x2)
)
(1− x2)
√
1− x2 + C, C ∈ R
4.16 x
(
ln3(2x)− 3 ln2(2x) + 6 ln(2x)− 6
)
+ C, C ∈ R
4.17 x arctg(ax)− 1
2a
ln (1 + a2x2) + C, C ∈ R
Página 9
4.18
(
3x2
32
− x
3
4
)
cos(4x) +
(
3x2
16
− 3
128
sen(4x)
)
+ C, C ∈ R
4.19 −xe−x + C, C ∈ R
4.20
x3
3
(
−1
3
+ lnx
)
+ C, C ∈ R
4.21 (x2 − 2x+ 2) ex + C, C ∈ R
4.22 x arcsen
(x
2
)
+
√
4− x2 + C, C ∈ R
4.23 (x− 1) tg x+ ln | cosx|+ C, C ∈ R
4.24
4e3x
25
(
sen(4x) +
3 cos(4x)
4
)
+ C, C ∈ R
4.25
(
lnx− 1
n+ 1
)
xn+1
n+ 1
+ C, C ∈ R
4.26 x ln(x2 + 1)− 2x+ 2 arctg x+ C, C ∈ R
4.27 x ln(x+
√
x2 + 1)−
√
x2 + 1 + C, C ∈ R
4.28
x2 arctg x
2
− x
2
+
arctg x
2
+ C, C ∈ R 4.29
(
x4
4
− x2 + 1
)
ex
2
+ C, C ∈ R
4.30
1
4
(
x2 + x sen(2x) +
1
2
cos(2x)
)
+ C, C ∈ R
4.31 (x2 + 4x+ 5) ex + C, C ∈ R
4.32
2x(x+ 1)
√
x+ 1
3
− 4(x+ 1)
2
√
x+ 1
15
+ C, C ∈ R
4.33
x (cos(lnx) + sen(ln x))
2
+ C, C ∈ R
4.34 x arccosx−
√
1− x2 + C, C ∈ R
4.35
secx tg x+ ln | secx+ tg x|
2
+ C, C ∈ R
4.36
(
1− 1
x
)
e
1
x + C, C ∈ R
5.
5.1
1
4
cos(2x)− 1
12
cos(6x) + C, C ∈ R
5.2
1
2
senx+
1
10
sen(5x) + C, C ∈ R
5.3
1
8
sen(4x)− 1
12
sen(6x) + C, C ∈ R
5.4 − 1
16
cos(8x) +
1
4
cos(2x) + C, C ∈ R
5.5 −sen(25x)
50
+
sen(5x)
10
+ C, C ∈ R
5.6
3
5
sen
(
5x
6
)
+ 3 sen
(x
6
)
+ C, C ∈ R
5.7
3
2
cos
(x
3
)
− 1
2
cosx+ C, C ∈ R
5.8
1
4a
sen(2ax) +
x
2
cos(2b) + C, C ∈ R
5.9
t
2
cos θ − 1
4ω
sen(2ωt+ θ) + C, C ∈ R
5.10 −(sen
2(2x)2 − 3) sen(2x)
6
+ C, C ∈ R
5.11
3
8
− sec
2(2x)
2
+
sec4(2x)
8
− ln | cos(2x)|
2
+ C,
C ∈ R
5.12 −x− 5 cotg
(x
5
)
+ C, C ∈ R
5.13
1
5
tg x sec4 x +
4
15
tg x sec2 x +
8
15
tg x + C,
C ∈ R
5.14 − cosx+ 2
3
cos3 x− 1
5
cos5 x+ C, C ∈ R
5.15 −4
3
sen
(x
4
)
sen2
(x
4
)
+ sen
(x
4
)
+C, C ∈ R
5.16
3x
8
+
1
4
sen(2x) +
1
32
sen(4x) + C, C ∈ R
5.17 −1
3
cotg3 x+ cotg x+ x+ C, C ∈ R
5.18 −1
3
cotg3 x− cotg x+ C, C ∈ R
Página 10
5.19 −1
2
tg2 x+ ln | cosx|+ C, C ∈ R
5.20 tg x+
1
3
tg3 x+ 2x− sen(2x) + C, C ∈ R
5.21 −1
3
cosec3 x+ cosecx+ C, C ∈ R
5.22
x
8
− 1
32
senx cosx+
1
4
sen3 x cosx+ C, C ∈ R
5.23
1
5
cos5 x− 1
3
cos3 x+ C, C ∈ R
5.24
x
16
− sen(2x)
24
+
sen(4x)
192
+
cosx sen5 x
6
+ C, C ∈ R
5.25
sec7 x
7
− 14 sec
5 x
35
+
sec3 x
3
+ C, C ∈ R
5.26
x
8
− 1
96
sen(12x) + C, C ∈ R
5.27
x
16
− sen(4x)− sen(8x)
128
− sen(12x)
384
+ C, C ∈ R
5.28 2 ln
∣∣∣cos(x
2
)∣∣∣+ tg2 (x
2
)
− 1
2
tg4
(x
2
)
+ tg
(x
2
)
+ C, C ∈ R
5.29 −cotg
3 x
3
− cotg
5 x
3
+ C, C ∈ R
5.30 −cotg
2 x
2
− ln | senx|+ C, C ∈ R
5.31 −1
4
cotg x cosec3 x− 3
8
cotg x cosecx− 3
8
ln | cosecx+ cotg x|+ C, C ∈ R
6.
6.1
9
2
arcsen
(x
3
)
+
x
√
9− x2
2
+ C, C ∈ R
6.2 ln
∣∣∣∣∣
√
4 + x2 + x
2
∣∣∣∣∣+ C, C ∈ R
6.3
1
2
ln
∣∣∣∣2−√4− x2x
∣∣∣∣+ C, C ∈ R
6.4
1
3
ln
∣∣∣∣∣
√
x2 + 9− 3
x
∣∣∣∣∣+ C, C ∈ R
6.5
√
x2 − 25
25x
+ C, C ∈ R
6.6 −
√
4− x2 + C, C ∈ R
6.7 − x√
x2 − 1
+ C, C ∈ R
6.8
1
432
(
arctg
(x
6
)
+
6x
x2 + 36
)
+ C, C ∈ R
6.9
1
2
arcsen
(
2x
3
)
+ C, C ∈ R
6.10
1
2(16− x2)
+ C, C ∈ R
6.11
√
(9x2 + 49)3
243
−
49
√
(9x2 + 49)3
81
+C, C ∈ R
6.12
(3 + 2x2)
√
x2 − 3
27x3
+ C, C ∈ R
6.13
arcsen(2x) + 2x
√
1− 4x2
4
+ C, C ∈ R
6.14
arcsenx− x
√
1− x2
2
+ C, C ∈ R
6.15
9
2
arcsen
(
x− 1
3
)
+
(x− 1)
√
9− (x− 1)2
2
+ C, C ∈ R
Página 11
6.16 −
√
1 + x2
x
+ C, C ∈ R
6.17 ln
(
2
√
x2 − 3x+ 2 + 2x− 3
)
+ C, C ∈ R
6.18 arcsen
(
x
3
− 2
3
)
+ C, C ∈ R
6.19
1
6
arctg
(
2x+ 2
3
)
+ C, C ∈ R
6.20
(x+ 1)
√
3− 2x− x2
2
+ 2 arcsen
(
x+ 1
2
)
+ C, C ∈ R
6.21
(x− 1)
√
x2 − 2x+ 2
2
+
ln(x− 1 +
√
x2 − 2x+ 2)
2
+ +C, C ∈ R
6.22
(2x+ 1)
√
x2 + x
8
−
ln
∣∣2x+ 1 + 2√x2 + x∣∣
8
+ C, C ∈ R
6.23
√
14
14
arctg
(√
14
7
(x− 1)
)
+ C, C ∈ R
6.24 x− ln
(
2 + ex + 2
√
e2x + ex + 1
)
+ C, C ∈ R
7.
7.1 ln |x3(x− 4)2|+ C, C ∈ R
7.2 ln(x− 1)6 + 5
x− 1
+ C, C ∈ R
7.3 ln
∣∣∣∣(x− 2)3(x+ 4)2
∣∣∣∣+ C, C ∈ R
7.4 ln
∣∣∣∣(x+ 1)5(x− 5)3
∣∣∣∣− 1x+ 1 + C, C ∈ R
7.5 ln
∣∣x5(x+ 3)4∣∣− 2
x
+
3
2x2
− 1
3x3
+ C, C ∈ R
7.6 x+ ln
[
x4(x2 + 4)
]
− 1
2
arctan
x
2
+ C, C ∈ R
7.7
1
2
x2 + x+ ln
[
x2(x− 1)2
]
+ C
7.8
1
3
x3 − 9x− 1
9x
− 1
2
ln(x2 + 9) +
728
27
arctan
x
3
+ C, C ∈ R
7.9 ln
[
(x− 4)2(x+ 1)2
]
− 3
2(x+ 1)2
+ C, C ∈ R
7.10
1
6
ln
∣∣(x− 3)(x+ 3)5∣∣− 7
2(x− 3)
− 3
2(x+ 3)
+ C, C ∈ R
7.11 ln
∣∣∣∣(x− 3)3(x− 2)2
∣∣∣∣+ C, C ∈ R
7.12 x+
1
4
ln
∣∣(x+ 1)(x− 3)19∣∣+ C, C ∈ R
Página 12
7.13 ln |x− 2| − 4
x− 2
− 5
2(x− 2)2
+ C, C ∈ R
7.14
x2
2
+
1
2
ln
∣∣∣∣(x− 3)3(x+ 1)x2
∣∣∣∣+ C, C ∈ R
7.15
1
3
ln
∣∣∣∣(x+ 1)(x− 2)5(x− 1)6
∣∣∣∣+ C, C ∈ R
7.16
1
x
+
2
x+ 1
+ 3 ln
∣∣∣∣ xx+ 1
∣∣∣∣+ C, C ∈ R
7.17
1
4
ln
∣∣∣∣(x− 2)(x+ 1)2(x+ 2)(x− 1)2
∣∣∣∣+ C, C ∈ R
7.18 ln
∣∣(x− 1)2(x2 + 6x+ 10)∣∣+ arctan(x+ 3) + C, C ∈ R
7.19
1
2
ln
∣∣∣∣(x+ 4)15(x+ 2)7
∣∣∣∣+ C, C ∈ R
7.20 − 5x+ 12
x2 + 6x+ 8
+ ln
(
x+ 4
x+ 2
)2
+ C, C ∈ R
7.21 ln
∣∣∣∣ x√x2 + 1
∣∣∣∣+ C, C ∈ R
7.22 ln
∣∣∣∣∣
√
(x2 − 2x+ 5)3
x− 1
∣∣∣∣∣+ 12 arctan
(
x− 1
2
)
+ C, C ∈ R
7.23 ln
(
x2 + 4√
x2 + 2
)
+
3
2
arctan
(x
2
)
− 3√
2
arctan
(
x√
2
)
+ C, C ∈ R
7.24 ln
(
x2 + 4
(x+ 1)2
)
+
1
2
arctan
(x
2
)
+ C, C ∈ R
7.25
1√
2
ln
∣∣∣∣∣x2 +
√
2x+ 1
x2 −
√
2x+ 1
∣∣∣∣∣+√2 arctan(√2x+ 1)+√2 arctan(√2x− 1)+ C, C ∈ R
7.26
1
3
(
x3 + ln
∣∣x3 − 1∣∣)+ C, C ∈ R
7.27
2− x
4(x2 + 2)
+ ln
(√
x2 + 2
)
− 1
4
√
2
arctan
(
x√
2
)
+ C, C ∈ R
7.28
3x2 − x
(x− 1)(x2 + 1)
+ ln
[
(x− 1)2
x2 + 1
]
+ arctanx+ C, C ∈ R
7.29 ln
∣∣∣∣x− 1x
∣∣∣∣− 103√3 arctan
(
2x− 1√
3
)
− 2x− 1
3(x2 − x+ 1)
+ C, C ∈ R
Página 13
Referências
[1] Diva Flemming e Mirian Gonçalves. Cálculo A, 6.a Edição. Pearson. São Paulo, 2006.
[2] B. Demidovitch e outros. Problemas e Exerćıcios de Análise Matemática, 5.a Edição. MIR. Mos-
covo, 1986.
[3] João Paulo Santos. Cálculo Numa Variável Real. IST Press. Lisboa, 2012.
[4] Ricardo Almeida e Rita Simões. Primitivas. Escolar Editora. Lisboa, 2014.
[5] Ana Castro, Ana Viamonte e António Sousa. Cálculo I - Conceitos, Exerćıcios e Aplicações.
Publindústria, Edições Técnicas. Porto, 2013.