Lista 5 - CM201 Cálculo I

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Universidade Federal do Parana´
5a Lista de Exerc´ıcios - CM201 Ca´lculo I
Professor: Cleber de Medeira 21/11/2013
1. Encontre o valor das integrais definidas.
(a)
∫ 3
0
(12x+ 1)dx
(b)
∫ 2
−1
(x3 + 2x2 − 3)dx
(c)
∫ e2
1
dx
x
(d)
∫ 2
−1
|x|dx
(e)
∫ 3
1
e2x+1dx
(f)
∫ 3
1
(x− 1)e(x2−2x)dx
(g)
∫ pi/2
0
x cos(x2 + 1)dx
(h)
∫ 2
−1
(x3 − 3x)dx
(i)
∫ 1
0
x4/5dx
(j)
∫ 2
1
3
t4
dt
(k)
∫ 9
1
x− 1√
x
dx
(l)
∫ pi/4
0
sec2tdt
(m)
∫ 2
1
(1 + 2y)2dy
(n)
∫ 2
0
t√
4− t2dt
2. Use o 1o Teorema Fundamental do Ca´lculo para encontrar a derivada das func¸o˜es.
(a) g(x) =
∫ x2
2
ln t dt
(b) g(x) =
∫ 1/x
2
arctg t dt
(c) g(x) =
∫ tg x
0
√
t+
√
t dt
(d) g(x) =
∫ 1
1−3x
u3
1 + u2
du
(e) g(x) =
∫ 3x
2x
u2 − 1
u2 + 1
du
(f) g(x) =
∫ 2
√
x
√
t sen tdt
3. Se F (x) =
∫ x
1
f(t)dt, sendo f(t) =
∫ t2
1
√
1 + u4
u
du encontre F ′′(2).
4. Se f(1) = 12, f ′ e´ cont´ınua e
∫ 4
1
f ′(x)dx = 17, qual e´ o valor de f(4)?
5. A func¸a˜o erro
err(x) =
2√
pi
∫ x
0
e−t
2
dt
e´ frequentemente usada em probabilidade, estat´ıstica e engenharia.
a) Mostre que
∫ b
a
e−t
2
dt =
√
pi
2
(err(b)− err(a)).
b) Mostre que a func¸a˜o y = ex
2
err(x) satisfaz a equac¸a˜o y′ = 2xy + 2/
√
pi.
6. Encontre uma func¸a˜o f e um nu´mero a tal que
6 +
∫ x
a
f(t)
t2
dt = 2
√
x, para todo x > 0.
7. Calcule as seguintes integrais indefinidas.
(a)
∫
(2x3 − 3x2 + 1)dx
(b)
∫
ex+1dx
(c)
∫
(1− x)(x2 + 1)dx
(d)
∫
x3 − 2√x
x
dx
(e)
∫
(1 + tg2x) senx dx
(f)
∫
1 + cos2 x
cos2 x
dx
(g)
∫
1 + 3
√
x√
x
dx
(h)
∫
t4 − 1
t2 − 1dt
(i)
∫
e−udu
(j)
∫
x2
√
x3 + 1dx
(k)
∫
senx cos3 x dx
(l)
∫
x sen(x2)dx
(m)
∫
dx
5− 3x
(n)
∫
(lnx)2
x
dx
(o)
∫
cos
√
t√
t
dt
(p)
∫
cos3 xdx
(q)
∫
ex
√
a+ exdx, a ∈ R
(r)
∫
z2
3
√
1 + z3
dz
(s)
∫
sen(2x)
1 + cos2 x
dx
(t)
∫
x+ 1
1− x2dx
8. (a) Se f e´ cont´ınua e
∫ 4
0 f(x)dx = 10 encontre
∫ 2
0 f(2x)dx.
(b) Se f e´ cont´ınua e
∫ 9
0 f(x)dx = 4 encontre
∫ 3
0 xf(x
2)dx.
(c) Se f e´ cont´ınua mostre que
∫ pi/2
0 f(cosx)dx =
∫ pi/2
0 f(senx)dx.
9. Calcule as integrais (Sugesta˜o: Use integrac¸a˜o por partes).
(a)
∫
x2 lnxdx
(b)
∫
x cosxdx
(c)
∫
x cos 3xdx
(d)
∫
xex/2dx
(e)
∫
x2 senxdx
(f)
∫
ln(2x+ 1)dx
(g)
∫
(lnx)2
(h)
∫
e2x senxdx
(i)
∫ pi
0 t sen tdt
(j)
∫ 2
1
lnx
x2
dx
10. Calcule a a´rea da regia˜o delimitada pelas curvas.
(a) y = x+ 2, y = x2
(b) y = lnx, x = 1, x = e
(c) y = x, y = x2
(d) y = x, y = x2, x = 0 e x = 2
(e) y = x2, y2 = x
(f) y = 12− x2, y = x2 − 6
(g) y =
√
x, x = 0 e x = 9
(h) x = 1− y2, x = y2 − 1
2
Algumas respostas
1. (a) 21/4
(b) 3/4
(c) 2
(d) 5/2
(e) 12(e
7 − e3)
(f) 12(e
3 − e−1)
(g) 12(sen(
pi2+4
4 ) −
sen 1)
(h) -3/4
(i) 5/9
(j) 7/8
(k) 40/3
(l) 1
(m) 49/3
(n) 2
2. (a) g′(x) = 2x lnx2
(b) g′(x) = − 1
x2
arctg 1x
(c) g′(x) = sec2 x
√
tg x
√
tg x
(d) g′(x) = 3(1−3x)
3
1+(1−3x)2
(e) g′(x) = −8x4−2
4x4+1
+ 27x
2−3
9x2+1
(f) g′(x) = − 14√x sen
√
x
3.
√
257 4. 29
7. (a)
x4
2
− x3 + x+ C
(b) ex+1 + C
(c) −x
4
4
+
x3
3
− x
2
2
+ x+ C
(d)
x3
3
− 4√x+ C
(e) secx+ C
(f) x+ tg x+ C
(g)
6
5
x5/6 + 2
√
x+ C
(h)
x3
3
+ x+ C
(i) −e−u + C
(j)
2
9
(x3 + 1)3/2 + C
(k) −1
4
cos4 x+ C
(l) −1
2
cosx2 + C
(m) −1
3
ln(5− 3x) + C
(n)
ln3(x)
3
+ C
(o) 2 sen
√
x+ C
(p)
1
12
(9 senx+ sen 3x) + C
(q)
2
3
(ex + a)3/2 + C
(r)
1
2
(x3 + 1)2/3 + C
(s) − ln(cos 2x+ 3) + C
(t) − ln(1− x) + C
8. (a) 5 (b) 2
9. (a)
1
9
x3(−1 + 3 lnx) + C
(b) x senx+ cosx+ C
(c)
1
9
(3x sen(3x) + cos(3x)) + C
(d) 2ex/2(x− 2) + C
(e) 2x senx− (x2 − 2) cosx+ C
(f)
1
2
(2x+ 1)(−1 + ln(2x+ 1)) + C
(g) x(2 + ln2 x− 2 lnx) + C
(h) −1
5
e2x(cosx− 2 senx) + C
(i) pi
(j)
1
2
(1− ln 2)
10. (a) 9/2
(b) 1
(c) 1/6
(d) 1
(e) 1/3
(f) 72
(g) 18
(h) 8/3
3