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1. Indeterminate Forms and L’Hospital’s Rule – Homework
Find the following limits.
⎛ 2x 2 − x − 3 ⎞
1. lim ⎜
x→−1 ⎝
x +1 ⎟⎠
⎛ x 3 − 5x −12 ⎞
2. lim ⎜
x→3 ⎝
x 2 − 3x ⎟⎠
⎛ x2 − x − 3⎞
3. lim ⎜
x→3 ⎝
x − 2 ⎟⎠
⎛
x −1 ⎞
4. lim ⎜
⎟
x→1 ⎝
x2 + 3 − 2 ⎠
x
⎛
⎞
5. lim ⎜
⎟
x→0 ⎝ 1 − x − e −2 x ⎠
⎛ x cos π x + sin x ⎞
6. lim ⎜
⎟⎠
x→0 ⎝
x
⎛ x 2 −1 ⎞
7. lim ⎜ 2
x→∞ ⎝ 4x + x ⎟
⎠
⎛ 2x 2 + 4x − 7 ⎞
8. lim ⎜ 3
x→−∞ ⎝ x + 3x 2 − 5 ⎟
⎠
⎛ x2 ⎞
9. lim ⎜ − x ⎟
x→−∞ ⎝ e
⎠
⎛ e x − e − x − 2x ⎞
11. lim ⎜
x→0 ⎝
x − sin x ⎟⎠
⎛ sin 2x − tan x − x ⎞
10. lim ⎜
⎟⎠
x→0 ⎝
x3
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π⎤
⎡
12. lim ⎢ x sin ⎥
x→∞ ⎣
x⎦
13. lim+ [ tan x ln x ]
14. lim [ csc x − cot x ]
15. lim ⎡⎣ xe1 x − x ⎤⎦
x→∞
1 ⎤
⎡ 1
−
16. lim ⎢
x→1 ⎣ ln x
x −1 ⎥⎦
2 ⎤
⎡ 1
− 2 ⎥
17. lim ⎢
x→0 ⎣1 − cos x
sin x ⎦
x→0
x→0
Limits of the type
⎡ x ⎤
18. lim+ ⎢
x→0 ⎣ ln x ⎥
⎦
0 ∞ ∞
, , 0 , ∞ + ∞, and ∞ ⋅ ∞ are not indeterminate. Find the following.
∞ 0
⎡1
⎤
19. lim+ ⎢ − ln x ⎥
x→0 ⎣ x
⎦
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20. lim − ( cos x )
x→(π 2 )
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tan x
⎛ 2
⎞
+ tan x ⎟
21. lim − ⎜
⎠
x→(π 2 ) ⎝ π − 2x
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2. Integration by Parts – Homework
Find the following integrals. Not all require integration by parts.
1.
∫ x sin x dx
2.
∫ x cos5x dx
3.
∫ xe
4.
∫ (6x + 2) e
5.
∫ x csc x dx
6.
∫
7.
∫
8.
8x
dx
2
2
xe x dx
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∫
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−3x
dx
x ln x dx
( ln x )4 dx
x
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∫
x 3 ln x dx
10.
∫
11.
∫x
x +1 dx
12.
∫ sin
13.
∫ e cos x dx
14.
∫e
9.
x
15. Find
4x
dx
e2 x
4x
−1
x dx
sin x dx
∫ x cos ( x ) dx
5
3
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16.
∫x e
4 x 2
dx
17.
e
18.
∫ 4x ln x dx
∫ x cos3x dx
3
1
19.
1
∫ x ln (1 + x ) dx
2
0
20. Find the volume when the region bounded by y = ln x, y = 0, and x = e is rotated about the x-axis.
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3. Integration by Partial Fractions – Homework
Find the following integrals:
∫
11x −15
dx
x 2 − 3x + 2
2.
∫
3x −12
dx
x − 5x − 50
5.
7.
∫
9x 2 − 25x − 50
dx
( x +1) ( x − 7) ( x + 2 )
9.
∫ x + 2x
1.
4.
2
3
2
x
dx
−16x − 32
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∫
7x + 25
dx
x − 7x − 8
3.
∫
5x −11
dx
x − 2x − 8
∫
21
dx
x + 7x +10
6.
∫
10x
dx
x − 9x − 36
2
2
8.
2
∫ x − x dx
10.
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2
1
3
4x + 6
dx
2
−1
∫ 4x
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11.
∫
x 3 +1
dx
x2 − x
12.
13. Use the substitution u = x +1
dx
to find
x x +1
14. Use the substitution u = x + 4
dx
to find
( x − 5) x + 4
∫
15. Find
∫
∫
cos x
dx
sin x (sin x −1)
17. Find the area bounded by y =
y = 0, x = −2, and x = 1.
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∫
x 3 − 7x 2 + 5x + 40
dx
x 2 − 2x − 8
16. Find
x+5
,
x + 7x +12
2
∫
sec 2 x
dx
tan x ( tan x +1)
25
from x = 2
x + 3x − 4
to x = b as b approaches infinity.
18. Find the area under y =
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2
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4. Improper Integrals – Homework
For each of the integrals below, explain whether or not it could converge based on the integrand. Then
determine mathematically whether or not it converges and if it can, find the limit to which it converges.
∞
1.
∫
4
∞
4.
∫
1
∞
7.
∫
0
1
dx
x2
1
dx
x 0.2
1
dx
x +1
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∞
2.
∫
e
∞
5.
∫
1
dx
x1.2
1
−1
dx
x ln x
1
8.
∫
0
∞
1
dx
x4
3.
∫ x 1
1
1
6.
∫x
1
0.2
dx
0
∞
9.
∫
x dx
0
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π 2
10.
∫
4
tan x dx
11.
0
0
9
13.
∫
0
∫
3
1
dx
x −1
2
14.
∫
0
∞
1
16 − x
2
dx
12.
∫
1
1
dx
( 2x −1)3
1
dx
x −x
2
11
15.
∫ ( x − 3)
−2 3
dx
2
Find the area of the unbounded shaded region:
16.
17.
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