CalA - 6 - 2 PDF
CalA - 6 - 2 PDF
CalA - 6 - 2 PDF
dx
x 2
1
+c= 2 +c
2
2x
d 1 1
=
dx 2 x 2 x 3
3
x dx =
2.
9 t
1
t3
t3 t 2
2
(
9
t
+
t
)
dt
=
9
.
+
+ c = 3t 3
+ c.
3 1
t
2
3
2
d
dt
3.
1 3
1
3 2
+ c = 9t 2 2. . t 2 = 9t 2 +
.
3t
2
t
t3
(ax
+ bx 3 + 3c)dx
x5
x4
= a + b + 3cx + C.
5
4
d a 5 b 4
a 4 b 3
4
3
x + x + 3cx + C = 5 x + 4 x + 3c = ax + bx + 3c.
dx 5
4
4
5
4.
x x
dx
3
443
x 2 1 x2
2 5
1 1 3
= x 2 + x 2 dx =
+ . + c = 2 x + x 2 + c.
1
3
3 5
15
2
2
d
2 5
1 1 2 5 3
1
x x
+
2 x + x2 + c = 2 x 2 + . .x2 =
dx
15
2
15 2
3
x
5.
(2 x
(4 x
3) 2 dx
12 x 2 + 9 dx = 4
x5
x3
4
12 + 9 x + c = x 5 4 x 3 + 9 x + c.
5
3
5
d 4 5
4 4
3
2
4
2
x 4 x + 9 x + c = 5 x 12 x + 9 + c = 4 x 12 x + 9 + c.
dx 5
5
6.
dx
sen
= sen 2 x dx
= cos sec 2 x dx = cot g x + c.
d
( cot g x + c ) = cos sec2 x = 1 2 .
dx
sen x
7.
2 y 1 dy
2 y
444
1
1 21
= 2 . y2
. y dy =
2
2 y2
1 y2
. +c
3
2 1
2
2
2 2 32
2 12
2
=
y
. y + c = 2 y y 1 + c.
3
2
3
2 2 32
2 2 3 12
1
2 12
2 1 12
1 12
2
y
.
y
c
.
y
.
y
2
y
y
3
3 2
2
2 2
2
1
= 2y
.
2y
d
dx
8.
3t
2 dt
2
=
arc tg t + c
+3
3
d 2
2 1
2
=
arc
tg
t
+
c
.
= 2
.
2
dt 3
3t + 3
3 1+ t
9.
x dx
9
x2
x
dx
=
+c
9
2
d 2 92
2 9 7
3
x + c = . x2 = x x
dx 9
9
2
7
2
10.
x5 + 2x 2 1
x 4 dx
= x + 2x
d
dx
x2
x 1 x 3
x2 2
1
dx =
+2
+c=
+ 3 +c
2
1 3
2 x 3x
x2 2
2 x 2 1 . 9 x2
1
2
1 x5 + 2 x 2 1
+ 3 + c =
2 +
=
x
+
=
.
x
x2 x4
x4
9 x6
2 x 3x
2
445
x2
x 2 + 1 dx
1
= 1 2
dx = x arc tg x + c.
x +1
12.
x2 +1
x 2 dx
1
x
1
(1 + x )dx = x + 1 + c = x x + c.
2
13.
sen x
dx
2
x
cos
=
14.
9
dx
1 x2
15.
3
1 x2
dx = 3arc sen x + c.
4
dx
x x2
x
16.
sen x
1
.
dx = tg x . sec x dx = sec x + c
cos x cos x
x2 1
dx = 2arc sec x + c.
8 x 4 9 x3 + 6 x2 2 x + 1
dx
x2
= 8 x 2 9 x + 6 2 x 1 + x 2 dx
=
8 x3 9 x2
1
+ 6 x 2 ln x + c
3
2
x
446
17.
et
1
2 + t + t dt
3
1
t2
1
2 3
= et + + ln t + c = et + t 2 + ln t + c.
3
2
2
3
2
18.
cos . tg d
= cos .
19.
(e
sen
d = send = cos + c.
cos
e x )dx
= 2 sen h x dx = 2 cosh x + c.
20.
(t +
t + 3 t + 4 t + 5 t )dt
3
t2 t 2 t 3 t 4 t 5
= + + + + +c
2 3 4 5 6
2 3 4 5
5
2
t
2 3 3 4 4
5 6
= + t2 + t3 + t4 + t5 + c
2 3
4
5
6
21.
x 1 / 3 5
x dx
4 5
= x 3 dx
x
22.
13
2
=
23.
5 ln | x | +c = 3 x
13
5 ln | x | + c.
2e t + cosh t )dt
2t
2 et + senh t + c.
ln 2
sec
x(cos 3 x + 1)dx
447
=
. cos 3 x + sec 2 x dx = senx + tgx + c.
2
cos x
24.
dx
, a 0, constante
2
+ a2
(ax)
dx
dx
1
= 2 2
= 2 arc tg x + c
2
a x +a
a ( x + 1) a
25.
2 2
x2 1
x 2 + 1 dx
2
= 1 2
dx = x 2arc tg x + c.
x +1
3
26.
1
8(t 2) t + dt
2
6
1
1
= 2 (t 2) 2 t + dt = 2 t 2 4t + 4 t + dt
2
2
1
7
= 2 t 3 + t 2 4t 2 2t + 4t + 2 dt = 2 t 3 t 2 + 2t + 2 dt
2
2
t4 7 t3
t2
t 4 7t 3
= 2 . + 2 + 2t + c =
+ 2 t 2 + 4t + c.
4
2
3
2
2
3
27.
4 16t +
3
dt
t3
t4
t 2
8 5 3
=e 2 +3
+ c = et t 4 t 2 + c.
5
2
5
2
4
t
28.
29.
ln x
x ln x
dx
ln x
1 dx
dx =
= ln | x | + c.
x 2 ln x
2 x
tg
x cos ec 2 x dx
sen 2 x 1
dx = sec 2 x dx = tg x + c.
2
2
cos x sen x
448
30.
( x 1) ( x + 1)
2
(
= (x
dx
)(
= x 2 2 x + 1 x 2 + 2 x + 1 dx
4
+ 2 x 3 + x 2 2 x 3 4 x 2 2 x + x 2 + 2 x + 1 dx
= x 4 2 x 2 + 1 dx =
31.
dt
, onde n z
1 n
n t
2
2dt = 2t + c
Se n = 0,
dt
2 dt
=
= 2 ln | t | +c
1 n
t
n t
2
Se n = 1,
Se n 1,
32.
x5
x3
2 + x + c.
5
3
n
2
n
t dt =
t 1 n
+c
1 (1 n )
n
2
x 3 x2
x + x dx =
+
+c
5
2
3
2
3
3 53 x 2
x +
+c
5
2
3 1
F (1) = + + c = 1
5 2
F ( x) =
3 1 10 6 5 1
c =1 =
=
5 2
10
10
F ( x) =
33.
3 53 x 2 1
x +
.
5
2 10
449
1
+ cos 2 x + c
2
d 2 1
1
f ( x)dx = x
34.
1
+ 1 que se anule no ponto x = 2.
x2
x 1
1
1
2
F ( x) = 2 + 1dx = x + 1 dx =
+x+c= +x+c
1
x
x
1
F ( 2) = + 2 + c
2
1
1 4 3
c= 2=
=
2
2
2
1
3
F ( x) = + x
x
2
35.
+ c, determinar f ( / 4).
d
1 2
1
2 2 1
f = sen 1 =
1 = .
=
4 4 2
2
4 4
4
36.
2 2
.
8
sen x dx = + cos x + c
f ( x) = + cos x + c
f (0) = cos 0 + c = 2
c = 2 1 = 1
f ( x) = cos x + 1.
450