Exercise on Integration1.1 Substitution Use a suitable substitution to evaluate the following integral... Z e2xcos 3xdx 1.3 Reduction Formula Prove the following reduction formulas... Z
Trang 1Exercise on Integration
1.1 Substitution
Use a suitable substitution to evaluate the following integral
1
Z
dx
√
2 − 5x
2
Z e3x+ 1
ex+ 1 dx
3
Z
x
√
1 − x2 dx
4
Z
x2√3
1 + x3dx
5
(1 + x2)2
6
√
x(1 + x)
7
Z
1
x2sin1
xdx 8
Z
xe−x2dx
9
Z
(ln x)2
x dx
10
Z
exdx
2 + ex
11
ex+ e−x
12
Z
cos√
x
√
x dx
13
Z tan xdx
14
Z dx
1 + ex
15
Z x(x2+ 2)99dx
16
Z
x
√
25 − x2dx 17
Z
x
√ 3x2+ 1dx 18
Z
x2
√
9 − x3dx 19
Z x(x + 2)99dx
20
Z xdx
√ 4x + 5 21
Z
x√
x − 1dx
22
Z (x + 2)√
x − 1dx
23
Z xdx
√
x + 9 24
Z
x3(1 + 3x2)12dx
1.2 Integration By Parts
1
Z
ln xdx
2
Z
x2ln xdx
3
Z ln x x
2
dx
4
Z
xe−xdx
Trang 2Z
x2e−2xdx
6
Z
x cos xdx
7
Z
x2sin 2xdx
8
Z
(ln x)2dx
9
Z
sin−1xdx
10
Z
x tan−1xdx
11
Z
ln(x +√
1 + x2)dx
12
Z
x sin2xdx
13
Z sin(ln x)dx
14
Z
x sin 4xdx
15
Z
x cos−1xdx
16
Z tan−1xdx
17
Z
x99ln xdx
18
Z
ln x
x101dx 19
Z
x sec2xdx
20
Z
e2xcos 3xdx
1.3 Reduction Formula
Prove the following reduction formulas
1 In=
Z
xneaxdx; In = x
neax
a − n
aIn−1, n ≥ 1
2 In=
Z
cosnxdx; In = sin x cos
n−1x
n − 1
n In−2, n ≥ 2
3 In=
Z
1 sinnxdx; In= −
cos x (n − 1) sinn−1x+
n − 2
n − 1In−2, n ≥ 2
4 In=
Z
xncos xdx; In = xnsin x + nxn−1cos x − n(n − 1)In−2, n ≥ 2
5 In=
(x2− a2)n; In = − x
2a2(n − 1)(x2− a2)n−1 + 2n − 3
2a2(n − 1)In−1, n ≥ 1
6 In=
Z
xndx
√
x + a; In =
2xn√
x + a 2n + 1 − 2an
2n + 1In−1, n ≥ 1
7 In=
Z
(ln x)ndx; In= x(ln x)n− nIn−1, n ≥ 1
8 In=
Z 1
0
xn√
1 − xdx; In = 2n
2n − 3In−1, n ≥ 2.
Trang 31.4 Trigonometric Integrals
Evaluate
1
1 − cos x
2
Z
sin5x cos xdx
3
Z
sin 3x sin 5xdx
4
Z
cosx
2cos
x
3dx 5
Z
cos3xdx
6
Z
sin4xdx
7
Z
sec2x tan2xdx
8
Z
sec x tan3xdx
9
Z
cot2xdx
10
cos x sin2x 11
Z sin x cos3x
1 + cos2x, dx 12
Z tan5xdx
13
sin4x cos4x, dx 14
Z sin 5x cos xdx
15
Z cos x cos 2x cos 3xdx
16
Z cos5x sin3xdx
17
Z cos5x sin4xdx
18
Z sin2x cos4xdx
1.5 Trigonometric Substitution
Evaluate the following integrals by trigonometric substitution
1
Z
x2
1 + x2 dx
2
(1 − x2)32
3
Z r 1 + x
1 − xdx
4
Z
dx
(1 + x2)3
5
Z x2dx
√
9 − x2
6
Z dx
√
4 + x2
7
Z
x2√
16 − x2dx
8
Z
dx
x2√
x2+ 4 9
(4x2+ 1)3/2 10
Z
1 (2x − x2)3/2
Trang 41.6 Rational Functions
Evaluate the following integrals of rational functions
1
Z
x2dx
1 − x2
2
Z x3
3 + xdx
3
Z (1 + x)2
1 + x2 dx
4
Z
dx
x2+ 2x − 3
5
(x2 − 2)(x2+ 3)
6
Z
x2+ 1 (x + 1)2(x − 1), dx
7
(x2 − 3x + 2)2, dx
8
Z
x2+ 5x + 4
x4+ 5x2+ 4, dx 9
Z
dx (x + 1)(x2+ 1) 10
Z 2x3− 4x2− x − 3
x2− 2x − 3 dx 11
Z
4 − 2x (x2+ 1)(x − 1)2 dx 12
Z
dx x(x2+ 1)2
13
(x − 1)(x − 2)(x − 3) 14
x2(x2− 2x + 2)
1.7 t-method
Use t-substitution to evaluate the following integrals
1
Z dx
sin3x
2
Z
dx
1 + sin x
3
sin x cos4x
4
2 + sin x 5
Z
1 − cos x
3 + cos xdx 6
Z cos x + 1 sin x + cos xdx
1.8 Miscellaneous
Evaluate the following integrals
1
Z (ln x)2
x dx
2
Z
x(ln x)2dx
3
Z
xdx
√
1 − x2
4
Z x + 4 (x + 1)2dx
5
Z cos3x sin2xdx
6
Z xdx (1 + x2)2
Trang 5Z e2xdx
1 + ex
8
x(1 + 2 ln x)
9
Z
cos2x sin3xdx
10
Z sin 2x
1 + cos2xdx
11
Z
e1x
x2dx
12
Z
sin x
cos2xdx
13
Z
x tan2xdx
14
Z
cot x
1 + sin xdx
15
Z x3dx
x2− 1
16
Z
dx
e2x+ ex− 2
17
Z
ln x
x√
1 + ln xdx
18
Z √
9 − x2
x2 dx
19
Z
x2dx
x2+ 1
20
√
x2+ 9
21
Z cos3x
sin x dx
22
Z
x2+ 8
x2− 5x + 6dx
23
Z xdx
√
x − 2
24
√
1 + ex
25
Z
cos(ln x)dx
26
Z
x sin2xdx
27
√
ex− 1 28
Z 4dx
x2√
4 − x2
29
Z
x + 1
x2(x − 1)dx 30
Z sec3x tan xdx
31
Z
x3√
x2 + 1dx
32
Z cos 2x sin 3xdx
33
Z x4+ x2− 1
x3+ x dx 34
Z x3dx
√
x2+ 4 35
(x2− 1)2
36
Z dx
1 +√ x 37
Z cos√ xdx
38
Z tan4xdx
39
Z
dx
√ x(x − 1) 40
Z
x2tan−1xdx
41
Z sin−1xdx
42
xdx
√
1 − x 43
Z √
x + 1
x dx 44
Z √
x√
1 − xdx
Trang 6Section 1.1: Substitution
1 −25√
2 − 5x + C
2 12e2x− ex+ x + C
3 −√
1 − x2 + C
4 14(1 + x3)4 + C
5 −2(1+x1 2 )+ C
6 2 tan−1√
x + C
7 cosx1 + C
8 −12e−x2 + C
9 13(ln x)3+ C
10 ln(2 + ex) + C
11 tan−1ex+ C
12 2 sin√
x + C
13 − ln | cos x| + C
14 x − ln(1 + ex) + C
15 2001 (x2+ 2)100+ C
16 −√
25 − x2+ C
17 1 3
√ 3x2+ 1 + C
18 −23√
9 − x3+ C
19 (x+2)101101 −(x+2)50100 + C
20 121 (2x − 5)√
4x + 5 + C
21 2
15(x − 1)3/2(3x + 2) + C
22 25(x − 1)3/2(x + 4) + C
23 23(x − 18)√
x + 9 + C
24 1
135(3x2+ 1)3/2(9x2 − 2) + C Section 1.2: Integration By Parts
1 x ln x − x + C
2 x33(ln x − 13) + C
3 −1x((ln x)2+ 2 ln x + 2) + C
4 −(x + 1)e−x+ C
5 −e−2x4 (2x2+ 2x + 1) + C
6 x sin x + cos x + C
7 −2x24−1cos 2x + x2 sin 2x + C
8 x(ln x)2− 2x ln x + 2x + C
9 x sin−1x +√
1 − x2+ C
10 −x2 +1+x22 tan−1x + C
11 x ln(x +√
1 + x2) −√
1 + x2+ C
12 x42 − x
4sin 2x − 18cos 2x + C
13 x2(sin(ln x) − cos(ln x)) + C
14 161 sin 4x − 14x cos 4x + C
15 x2cos2−1x +sin−14 x −x√1−x 2
16 x tan−1x − 12log(x2+ 1) + C
17 1
100x100ln x − x100
10000 + C
18 −10000x1 100 − ln x
100x 100 + C
19 x tan x + ln(cos x) + C
20 131 e2x(3 sin 3x + 2 cos 3x) + C
Trang 7Section 1.4: Trigonometric Integrals
1 − cotx2 + C
2 16sin6x + C
3 14sin 2x − 161 sin 8x + C
4 3 sinx6 +35 sin5x6 + C
5 sin x − 13sin3x + C
6 3
8x − 1
4sin 2x + 1
32sin 4x + C
7 13tan3x + C
8 13sec3x + − sec x + C
9 −x − cot x + C
10 −sin x1 + 12ln1+sin x1−sin x + C
11 −12cos2x +12 ln(1 + cos2x) + C
12 tan44 −tan 2 x
2 − ln | cos x| + C
13 −8 cot 2x − 83cot32x + C
14 −18cos 4x − 121 cos 6x + C
15 x4 + sin 2x8 +sin 4x16 + sin 6x24 + C
16 cos88(x)− cos66(x) + C
17 sin99(x) − 2 sin77(x) +sin55(x) + C
18 −16cos5x sin x + 241 cos3x sin x +
1
16cos x sin x +161 x + C
Section 1.5: Trigonometric Substitution
1 x − tan−1x + C
2 √x
1−x 2 + C
3 −√
1 − x2 + sin−1x + C
4 √x
1+x 2 + C
5 92sin−1 x3 − x
2
√
9 − x2+ C
6 ln |x +√
4 + x2| + C
7 √
16 − x2x 3
4 − 2x+32 sin−1 x4 +C
8 −
√
x 2 +4
9 √ x 4x 2 +1
10 √x−1 2x−x 2
Section 1.6: Rational Functions
1 −x + 1
2ln |1+x
1−x| + C
2 9x − 32x2 +13x3− 27 ln |3 + x| + C
3 x + ln(1 + x2) + C
4 14ln |x−1x+3| + C
5 1
10√2ln |x−
√ 2 x+√2| − 1
5√3tan−1 x√
3 + C
6 x+11 +12ln |x2− 1| + C
7 −x25x−6−3x+2+ 4 ln |x−1x−2| + C
8 tan−1x +5
6lnx2+1
x 2 +4 + C
9 12tan−1x +14ln(x+1)x2 +12 + C
10 x2+ 2 ln |x + 1| + 3 ln |x − 3| + C
11 tan−1x − 1
x−1 + ln x2+1
(x−1) 2 + C
12 2(x21+1) + ln |x| − 12ln(x2 + 1) + C
13 92ln(x−3)−4 ln(x−2)+12ln(x−1)+C
14 14lnx2 −2x+2x2
− 1
2tan−1(1 − x) + C
Trang 8Section 1.7: t-method
1 −2 sincos x2 x +12 ln | tanx2| + C
2 tan x − sec x + C
3 cos x1 +3 cos13 x + ln | tanx2| + C
4 √2
3tan−1
2 tan(x
2)+1
√ 3
+ C
5 2√
2 tan−1
tan(x
2)
√ 2
− x + C
6 12(x + ln(sin x + cos x + 3)) −√1
7tan−1
2 tan(x
2)+1
√ 7
+ C
Section 1.8: Miscellaneous
1 13(ln x)3+ C
2 12x2(ln x)2− 1
2x2ln x + 14x2+ C
3 −√
1 − x2 + C
4 ln |x + 1| − x+13 + C
5 −sin x1 − sin x + C
6 −2(1+x1 2 )+ C
7 ex− ln(1 + ex) + C
8 12ln |1 + 2 ln x| + C
9 15cos5x − 13cos3x + C
10 − ln(1 + cos2x) + C
11 −e1x + C
12 sec x + C
13 −x22 + x tan x + ln cos x + C
14 − ln |1 + csc x| + C
15 12x2+12ln |x2− 1| + C
16 −x
2 +
1
3ln |ex− 1| + C
17 −4
3
√
1 + ln x +2
3(ln x)√
1 + ln x + C
18 −
√
9−x 2
x − sin−1 x
19 x − tan−1x + C
20 ln |x +√
x2+ 9| + C
21 ln | sin x| − 12sin2x + C
22 x + 17 ln |x − 3| − 12 ln |x − 2| + C
23 23(x − 2)32 + 4(x − 2)12 + C
24 x − 2 ln(1 +√
1 + ex) + C
25 x2(cos(ln x) + sin(ln x)) + C
26 14x2 −1
4x sin 2x − 18 cos 2x + C
27 −2 sin−1e−x2 + C
28 −
√ 4−x 2
29 1
x − 2 ln |x| + 2 ln |x − 1| + C
30 13sec3x + C
31 13x2(x2+ 1)32 − 2
15(x2+ 1)52 + C
32 −101 cos 5x − 12cos x + C
33 12x2 − ln |x| +1
2 ln(x2+ 1) + C
34 13(x2+ 4)3 − 4√x2+ C
35 1
4ln |x + 1| − 1
4ln |x − 1| − x
2(x 2 −1) + C
36 2√
x − 2 ln(1 +√
x) + C
37 2√
x sin√
x + 2 cos√
x + C
38 13tan3x − tan x + x + C
39 ln |√
x − 1| − ln |√
x + 1| + C
40 13x3tan−1x − 16x2+16ln(x2+ 1) + C
41 x sin−1x +√
1 − x2+ C
42 sin−1√
x −√
x√
1 − x + C
43 2√
x + 1 + ln |√
x − 1| − ln |√
x + 1| + C
44 14sin−1√
x −14√
x√
1 − x(1 − 2x) + C