Engenharia ambiental
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ALUNO (a): 3a Lista de Exerc´ ıcios 1. Calcule as integrais definidas:5 2
. PER´ IODO: TURNO:
. SEMESTRE: 2012-2 . .
´ DISCIPLINA: CALCULO DIFERENCIAL E INTEGRAL II.
(a)
2 2
3 dx 6 dx
−1
(c) (d)
5
√
4
5 dx
(e)
1
√ √ 3 ( 2t + t)dt
5
(g)
1 1 0
√
2t − 1dt
2
−2 −2
π 2
(b)
2 dx
(f)
0
cos (x) dx (h) [1 + sen (x)]3
2
xex
−1
dx
2
2. Calcule as integrais definidas usando os seguintes resultados:
−1 π 2 π π
x2 dx = 3,
−1
dx =
3 , 2
sen (x) dx = 2,
0 2 −1
x dx = 3,
0
2
cos (x)dx = 0,
0 −2
π sen (x)dx = . 2
2
(a)
−1 2
(2x2 − 4x + 5)dx 2 − 5x +
−1 −1
(d)
−1 π
(x − 1)(2x + 3)dx (2 sen (x) + 3 cos (x) + 1)dx
0 π
(b) (c)
2
x2 2
dx
(e) (f)
0
(2x + 1)2 dx
( cos (x) + 4)2 dx
3. Calcule as seguintes integrais:
3
(a)
−1 2
4dx (h)
3
−1 −2 1
(b)
0 3
(x + 3x − 1)dx (i) (3x2 − 4x + 1)dx
0 6
1 +x x2
1
dx
(n)
0 0
√ 3
5 − xdx
e2x dx
−1 4
(o)
−1 1
√ x x + 1dx
(c) (d)
3 5
(j) (x2 − 2x)dx (k) |x − 3|dx
0 1
(5x +
1
√
x)dx
(e)
−2
π 8
1 dx x+1 sec 2 xdx
π 4
(l) sen (2x)dx
0
(f)
0 2
1 dx (x + 1)5 0 1 x2 (q) dx 3 0 1+x 1 x2 (r) dx 3 2 0 (1 + x ) (p) π 3
2
(g)
1
1 dx x2
(m)
1
(x − 2)5 dx
(s)
0
cos (2x)dx
4. Calcule as seguintes integrais: π 2
1
(a) π 4
cos (x) ln ( sen (x)) dx (e) x5 dx x 1
0 1 0 2
(b)
0
(f)
√ 3
ekt dt , k = 0 k x dx 5 (x2 + 4) √ x2 dx + 4x + 8 9 − e2t dt
a
(i)
0 1
x
a2 − x2 dx a2 + x2 x2
π
(c)
0
π 3
x5 cos x3 dx tg (x) sec 3 (x) dx
(g)
0
ln (3)
(d)
0
(h)
0
et
3 dx (x + 1) 2 dx (k) dx 2 + 12x − 7 1 4x 4 2x2 + 1 (l) dx 2 2 (x + 1) (x + 2) 0
(j)
4
2
5. Se f for cont´ ınua e
0 9
f (x) dx = 10, encontre
0 3
f (2x) dx. xf x2 dx.
0
6. Se f for cont´ ınua e
0
f (x) dx = 4, encontre
7. Se f for cont´ ınua em R, demonstre