Calculus II > Mixed Integration Practice > Flashcards

Mixed Integration Practice Flashcards

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1
Q

Integrate x * e^(-x) dx.

A

-xe^(-x) - e^(-x) + C (IBP)

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2
Q

Integrate dx / (x^2 + 2x + 10).

A

(1/3)arctan((x+1)/3) + C (Comp. Square)

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3
Q

Integrate sin^3(x) dx.

A

(1/3)cos^3(x) - cos(x) + C (Trig Integral)

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4
Q

Integrate dx / (x^2 - 9).

A

(1/6)ln|(x-3)/(x+3)| + C (Partial Fractions)

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5
Q

Integrate x / sqrt(1 - x^2) dx.

A

-sqrt(1 - x^2) + C (U-Sub)

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6
Q

Integrate ln(x) dx.

A

xln(x) - x + C (IBP)

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7
Q

Integrate sqrt(4 - x^2) dx.

A

2arcsin(x/2) + (x/2)sqrt(4-x^2) + C (Trig Sub)

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8
Q

Integrate (x+1) / (x^2 + x) dx.

A

ln|x| + C (Simplify first!)

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9
Q

Integrate x * cos(x^2) dx.

A

(1/2)sin(x^2) + C (U-Sub)

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10
Q

Integrate dx / (x^2 + 1)^2.

A

(1/2)arctan(x) + x/(2(x^2+1)) + C (Trig Sub)

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11
Q

Integrate tan^2(x)sec^2(x) dx.

A

(1/3)tan^3(x) + C (U-Sub)

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12
Q

Integrate x^2 * ln(x) dx.

A

(1/3)x^3ln(x) - (1/9)x^3 + C (IBP)

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13
Q

Integrate dx / sqrt(x^2 + 1).

A

ln|x + sqrt(x^2+1)| + C (Trig Sub)

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14
Q

Integrate 1 / (x^2 + 3x + 2) dx.

A

ln|(x+1)/(x+2)| + C (Partial Fractions)

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15
Q

Integrate e^sqrt(x) dx.

A

2(sqrt(x)-1)e^sqrt(x) + C (U-Sub then IBP)

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16
Q

Integrate cos^2(x) dx.

A

(1/2)x + (1/4)sin(2x) + C (Trig Identity)

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17
Q

Integrate x / (x+1) dx.

A

x - ln|x+1| + C (Long division)

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18
Q

Integrate dx / (x * ln x).

A

ln|ln x| + C (U-Sub)

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19
Q

Integrate x^2 / sqrt(x^3 + 1) dx.

A

(2/3)sqrt(x^3+1) + C (U-Sub)

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20
Q

Integrate sin(ln x) dx.

A

(x/2)(sin(ln x) - cos(ln x)) + C (IBP Loop)

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21
Q

Integrate dx / (x^2 + 4).

A

(1/2)arctan(x/2) + C (Trig Sub)

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22
Q

Integrate x^3 / (x^2 + 1) dx.

A

(1/2)x^2 - (1/2)ln(x^2+1) + C (Long division)

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23
Q

Integrate sec(x) dx.

A

ln|sec x + tan x| + C (Formula)

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24
Q

Integrate (ln x) / x dx.

A

(1/2)(ln x)^2 + C (U-Sub)

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25
Integrate dx / (x^2 - 1).
(1/2)ln|(x-1)/(x+1)| + C (Partial Fractions)
26
Integrate x * sqrt(x-1) dx.
(2/5)(x-1)^(5/2) + (2/3)(x-1)^(3/2) + C (U-Sub)
27
Integrate tan(x) dx.
ln|sec x| + C (Formula)
28
Integrate dx / sqrt(9 - x^2).
arcsin(x/3) + C (Trig Sub)
29
Integrate x * sec^2(x) dx.
xtan(x) + ln|cos x| + C (IBP)
30
Integrate 1 / (x^2 + x) dx.
ln|x/(x+1)| + C (Partial Fractions)
31
Integrate e^x * cos(x) dx.
(1/2)e^x(sin x + cos x) + C (IBP Loop)
32
Integrate dx / (x * sqrt(x^2 - 4)).
(1/2)arcsec|x/2| + C (Trig Sub)
33
Integrate sin^2(x)cos^2(x) dx.
(1/8)x - (1/32)sin(4x) + C (Trig Identity)
34
Integrate x^2 * e^x dx.
e^x(x^2 - 2x + 2) + C (IBP twice)
35
Integrate dx / (x^2 + 4x + 5).
arctan(x+2) + C (Comp. Square)
36
Integrate sqrt(x^2 - 1) dx.
(1/2)(x*sqrt(x^2-1) - ln|x + sqrt(x^2-1)|) + C (Trig Sub)
37
Integrate (x^2 + 1) / x dx.
(1/2)x^2 + ln|x| + C (Simplify)
38
Integrate dx / (x^2 - 4).
(1/4)ln|(x-2)/(x+2)| + C (Partial Fractions)
39
Integrate arctan(x) dx.
xarctan(x) - (1/2)ln(1+x^2) + C (IBP)
40
Integrate x / (x^2 + 1) dx.
(1/2)ln(x^2+1) + C (U-Sub)
41
Integrate sin(x)cos(x) dx.
(1/2)sin^2(x) + C (U-Sub)
42
Integrate dx / x^2 * sqrt(x^2 + 1).
-sqrt(x^2+1)/x + C (Trig Sub)
43
Integrate e^(2x) dx.
(1/2)e^(2x) + C (U-Sub)
44
Integrate (3x-1) / (x^2 - 1) dx.
ln|x-1| + 2ln|x+1| + C (Partial Fractions)
45
Integrate x^3 * ln(x) dx.
(1/4)x^4ln(x) - (1/16)x^4 + C (IBP)
46
Integrate dx / sqrt(1 - 4x^2).
(1/2)arcsin(2x) + C (U-Sub)
47
Integrate tan^3(x) dx.
(1/2)tan^2(x) + ln|cos x| + C (Trig Identity)
48
Integrate x * (x^2 + 1)^5 dx.
(1/12)(x^2 + 1)^6 + C (U-Sub)
49
Integrate dx / (x^2 + 6x + 13).
(1/2)arctan((x+3)/2) + C (Comp. Square)
50
Integrate sqrt(1 - x^2) dx.
(1/2)(arcsin x + x*sqrt(1-x^2)) + C (Trig Sub)
51
Integrate 1 / (x^3 + x) dx.
ln|x| - (1/2)ln(x^2+1) + C (Partial Fractions)
52
Integrate x^2 * sin(x) dx.
-x^2cos x + 2xsin x + 2cos x + C (IBP twice)
53
Integrate dx / x * sqrt(x^2 + 4).
-(1/2)ln|(2 + sqrt(x^2+4))/x| + C (Trig Sub)
54
Integrate sin^4(x) dx.
(3/8)x - (1/4)sin(2x) + (1/32)sin(4x) + C (Trig Identity)
55
Integrate e^x / (e^x + 1) dx.
ln(e^x + 1) + C (U-Sub)
56
Integrate (x+2) / (x-1) dx.
x + 3ln|x-1| + C (Long division)
57
Integrate dx / (x^2 + 2x + 2).
arctan(x+1) + C (Comp. Square)
58
Integrate x / (x^2 - 1) dx.
(1/2)ln|x^2 - 1| + C (U-Sub)
59
Integrate sin(2x) dx.
-(1/2)cos(2x) + C (U-Sub)
60
Integrate dx / (x^2 * sqrt(9 - x^2)).
-sqrt(9-x^2)/(9x) + C (Trig Sub)
61
Integrate x * e^(x^2) dx.
(1/2)e^(x^2) + C (U-Sub)
62
Integrate (2x+1) / (x^2+x) dx.
ln|x^2+x| + C (U-Sub)
63
Integrate dx / (x^2 + 4x + 8).
(1/2)arctan((x+2)/2) + C (Comp. Square)
64
Integrate sqrt(x^2 + 9) dx.
(1/2)(x*sqrt(x^2+9) + 9ln|x + sqrt(x^2+9)|) + C (Trig Sub)
65
Integrate 1 / (x^2 - x) dx.
ln|(x-1)/x| + C (Partial Fractions)
66
Integrate x * sin(x) dx.
sin x - xcos x + C (IBP)
67
Integrate dx / sqrt(x^2 - 4).
ln|x + sqrt(x^2-4)| + C (Trig Sub)
68
Integrate tan^4(x) dx.
(1/3)tan^3(x) - tan x + x + C (Trig Identity)
69
Integrate e^x * (1 + e^x)^2 dx.
(1/3)(1 + e^x)^3 + C (U-Sub)
70
Integrate (x-1) / (x+1) dx.
x - 2ln|x+1| + C (Long division)
71
Integrate dx / (x^2 + 2x + 5).
(1/2)arctan((x+1)/2) + C (Comp. Square)
72
Integrate x / sqrt(x+1) dx.
(2/3)(x+1)^(3/2) - 2sqrt(x+1) + C (U-Sub)
73
Integrate sin^2(x) dx.
(1/2)x - (1/4)sin(2x) + C (Trig Identity)
74
Integrate dx / (x^2 * sqrt(x^2 - 4)).
sqrt(x^2-4)/(4x) + C (Trig Sub)
75
Integrate x^2 * e^(-x) dx.
-e^(-x)(x^2 + 2x + 2) + C (IBP twice)
76
Integrate (x+3) / (x^2 + 3x + 2) dx.
2ln|x+1| - ln|x+2| + C (Partial Fractions)
77
Integrate dx / sqrt(4x - x^2).
arcsin((x-2)/2) + C (Comp. Square)
78
Integrate x * sec^2(x^2) dx.
(1/2)tan(x^2) + C (U-Sub)
79
Integrate (ln x)^2 / x dx.
(1/3)(ln x)^3 + C (U-Sub)
80
Integrate dx / (x^2 + 1)^(3/2).
x / sqrt(x^2+1) + C (Trig Sub)
81
Integrate e^(3x) * sin(x) dx.
(1/10)e^(3x)(3sin x - cos x) + C (IBP Loop)
82
Integrate (x^2+x+1) / (x^2+1) dx.
x + (1/2)ln(x^2+1) + C (Long division)
83
Integrate dx / (x * (ln x)^2).
-1/(ln x) + C (U-Sub)
84
Integrate sin^5(x) dx.
-cos x + (2/3)cos^3 x - (1/5)cos^5 x + C (Trig Integral)
85
Integrate x / (x^2 + 4x + 5) dx.
(1/2)ln(x^2+4x+5) - 2arctan(x+2) + C (U-Sub + Comp. Square)
86
Integrate dx / (x^2 + 9)^(2).
(1/54)arctan(x/3) + x/(18(x^2+9)) + C (Trig Sub)
87
Integrate e^x * cos(e^x) dx.
sin(e^x) + C (U-Sub)
88
Integrate (x^2) / (x-1) dx.
(1/2)x^2 + x + ln|x-1| + C (Long division)
89
Integrate dx / (x^2 - 2x + 2).
arctan(x-1) + C (Comp. Square)
90
Integrate x^3 * sqrt(1 - x^2) dx.
(1/5)(1-x^2)^(5/2) - (1/3)(1-x^2)^(3/2) + C (U-Sub)
Calculus II
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