Hơn 200 bài tập tích phân thông dụng cho ôn thi ĐH Toán 2014, các dạng bài tập tích phân, đề thi tích phân các năm đại học, hướng dẫn giải, phương pháp phân tích dạng tích phân thi đại học, tích phân cơ bản,
Trang 1Câu 1 Tính các tích phân xác định sau:
Tính các tích phân sau bằng đổi biến
1 R1
0(x4
1(x2
− x + 1x −x12)dx
3 R3
0
√x + 1dx
5 R
π 2 π 6
(cos x −√3 sin x + 1
x)dx 6 R1
0 x(ex 2
+ x3
)dx
7 R1
0(x2
+ x√x
1(√x + 1)(x −√x+ 1)dx
9 R2
−1
x
1
7x − 2√x− 5
11 R6 2
dx
√x + 2 −√x− 2 12 Re 2
e
(x + 1)dx (x2+ x) ln x
13 R
π 2 π 6
0
ex
− e− x
ex
+ e− x
dx
15 Rln 3 0
r
ex
ex
+ e− x
1
dx
√ 4x2 + 8x
17 R1 0
dx
ex
π 2
0
dx
1 + sin x
19 R
1
√e 1 e
dx
π 2 π 3 sin3xcos2xdx
21 R
π 2 π 3 sin2xcos3xdx 22 R
π 2
0
cos x
2 − 3 sin xdx
23 R
π 2
0
sin 2x
2 − 3 cos2xdx 24 R
π 6
0
√
3 + 4 sin x cos xdx
25 R1
0 x√
0 xex 2 +1dx
Trang 227 R1 0
√
0 x3√
x2+ 1dx
29 R2 1
dx
x√
1
dx
√
x2+ 3
31 R1 0
dx (1 + 3x2)2 32 R
π 4
0 esin 2
x
sin 2xdx
33 R1
0 xex 2 +2dx 34 R
π 3 π 4 sin3xcos2xdx
35 R
π 4
1
3
√
1 + 2 ln x
37 Re 1
√
3 ln x + 2 ln x
1
e3 ln x+1
x dx
39 Re 2
e
1 + ln3x
xln x dx 40 Re 2
e
dx
xcos2(1 + ln x)
41 R2 1
x
1 +√x
1
√
x+ 1
x dx
43 R
π 2
0 (sin4x+ 1) cos xdx 44 R1
0
√
4 − x2dx
45 R1 0
4x + 11
x2+ 5x + 6dx 46 R
π 6
0 (sin6x+ cos6x)dx
47 R
π 2
0
4 sin3x
π 4
0
1 + sin 2x cos2x dx
49 R
π 2
π 4
0
sin 4x
1 + cos2xdx
51 R
π 4
0
dx
π 2
0 sin 2x(1 + sin2x)dx
53 R
π 4
0
dx
π 4 π 6 dx sin x
Trang 355 R 3
0
tan4x
0 (1 − tan8x)dx
57 R
π 3 π 4
ln(tan x)
π 2 π 4
sin x − cos x
3
√
1 + sin 2xdx
59 R
π 2
0
sin 2x + sin x
√
1 + 3 cos xdx 60 R
π 2
0
sin3x
1 + cos2xdx
61 R
π 3 π 6
dx
π 2
0
cos3x
1 + cos xdx
63 R
π 2
0
dx sin x + cos x + 1 64 R
π 2
0
2 sin x − 3 cos x + 3 sin x + cos x + 1 dx
65 R
π 2 π 4
dx
π 3 π 4
3
p sin3x− sin x sin3xtan x dx
67 R
π 4
0 (cos4x+ sin4x) cos xdx 68 R
π 3 π 6 sin 2x sin 7xdx
69 R
π 3 π 4
sin2x
π 3
0
4 sin x (sin x + cos x)3dx
71 R
π 3 π 6
sin x + cos x
5
√ sin x − cos xdx 72 R
π 4
0
tan x
√
1 + sin2x
dx
73 R
π 4
π 3 π 6
cos2xdx sin2x+ 4 sin x cos x
75 R
π 2
0
sin2014x sin2014x+ cos2014xdx 76 R
π 2
0
dx
1 + sin 2x
77 R
π 2
0
dx
√
3 sin 2x + cos 2x 78 R1
0
xetan x 2
(etan x 2
+ 1) cos2x2dx
Trang 479 R
π 4
0
sin3x
π 4
0 (1 + sin2x)4
sin 2xdx
81 R
π 2
0 | cos x|√sin xdx 82 R
π 2
0
dx
2 sin x + 1
83 R
π 2 π 4 cos3xsin5xdx 84 R
π 4
0
sin 4x (1 + cos2x) sin2xdx
85 R
π 3 π 6
dx cos x sin5x 86 Rπ
− π
√
1 − sin xdx
87 R
π 4
0
dx (sin x + 2 cos x)2 88 R
π 2
0 esin 2
x
sin x cos3xdx
89 R2√3
√ 5
dx
x√x2
1 2
−
1 2
dx (2x + 3)√
4x2+ 12x + 5
91 R2 1
dx
x√
1
√
x2 + 2014dx
93 R2 1
dx
√
x2+ 2014 94 R1
0 x2√
x2+ 1dx
95 R3 0
√
9 − x2dx 96 R1
0 p(1 − x2)3dx
97 R√3 1
x2+ 1
x2√
x2+ 1dx 98 R
√ 2 2
0
r 1 + x
1 − xdx
99 R1 0
dx p(x2+ 1)3 100 R
√ 2 2
0
dx p(1 − x2)3
101 R
√ 2 2
0
x2dx
√
1 − x2 102 R√7
0
x3
3
√
1 + x2dx
103 R3
0 x3√
10 − x2dx 104 R1
0
xdx
√ 2x + 1
Trang 5105 R7 2
dx
√ 2x + 1 + 1 106 R1
0
x dx
x+√
x2+ 1
107 R√3 0
x5
+ x3
√x2
0
√
x3
− 2x2+ xdx
109 Re 1
ln3xp2 + 3 ln2x
ln 2
ln2x
x√
ln x + 1dx
111 R7
0 x(ex
+√3x
+ 1)dx 112 Rln 2
0
ex
dx p(ex
+ 1)3
113 R5
−2
x+ 2
3
√x
1
1 +√x
1 +√3xdx
115 R√7 0
x3
√
9 + x2dx 116 R1
0
dx (1 + x2)3
117 R1 0
(x + 1)dx
√
x2+ x + 1 118 R2
1
dx
x(x + 1)2
119 R1 0
dx (x + 1)(x2+ 1) 120 R1
0
x+ 1
x3
− 7x + 6dx
121 R1 0
1 − x4
0
1 +√4x
1 +√xdx
123 R1 0
r 1 − x
1 + x
dx
1
x√3 x
+ 2
x+√3x
+ 2dx
125 R2 1
xdx
4
px3(x + 1) 126 R16
1
dx (1 +√4x)√x
127 R
1
4
√ 2
0
dx
√
−1
xdx
√
5 − 4x
129 R1 0
r
ex
ex
+ e− x
0
√ex
− 1dx
131 R0
− ln 2
ex√
1 − e2xdx 132 R√3−1
−1
dx
x2+ 2x + 2
133 R2 1
1 + x2
1 + x4dx 134 R√3
1
√
1 + x2
x2 dx
Trang 6135 R2 1
1 − x2
1 + x4dx 136 R1
0
1
9 − x2 ln3 + x
3 − xdx
137 R
π 6
0
tan4x cos 2xdx 138 R1
0
dx
4 − x2
139 R1 0
x3
(1 + x2)3dx 140 R2
0 x2√
4 − x2dx
141 R1
0 x5
(1 − x3
)6dx 142 R3
0
x2
+ 1
√x + 1dx
143 R1 0
4x
x4+ 1dx 144 R2
1
√
x2+ 1 +√3
x3+ 1
x√
x2 + 1√3
x3+ 1 Tính các tích phân sau bằng phương pháp tích phân từng phần
145 Re 1
ln3x
1 xln xdx
147 R1
0 xln(x2
+ 1)dx 148 Re
1(x3
+ 2) ln xdx
149 R4
3 ln(x3
− 7x + 6)dx 150 R1
0 x2e3xdx
151 R1
0 xln(3 + x2
1(x2
− x) ln xdx
153 Re
1 xln2xdx 154 R2
1
ln(1 + x)
x3 dx
155 R1
0(x + 1)2e3xdx 156 Re
1(x ln x)2dx
157 Re
1 e
ln x (x + 1)2dx 158 R1
0 xln(x2
+ 1)dx
159 Re 1
ln x
0(2x + 7) ln(x + 1)dx
161 R1
−1
e−2 xx2dx 162 R1
0 ln(x +√
x2+ 1dx)
163 R1 0
(x2
+ 1)ex
(x + 1)2 dx 164 R1
0 ln(ex
+ 1)
Trang 7165 Rπ3 6
0 ex
cos xdx
167 R
π 6
π 2
0 (x2
+ 2x) sin xdx
169 R
π 3
0
x
π 2
0 x2
cos2xdx
171 R1
0 e2x
0 sin√xdx
173 R
π 3
0
x+ sin x
π 4
0 x2
(cos3x+ cos x)dx
175 R
π 2
0 cos x ln(1 + cos x)dx 176 R1
0(x tan x)2dx
177 R
π 4
0
xtan3
π 3
0 e3x
sin 4xdx
179 R
π 2
0 (x3
+ 2x)(sin3x+ sin 3x)dx 180 R
π 3
0 x3
cos xdx
Trang 8MỘT SỐ BÀI TÍCH PHÂN TRONG CÁC ĐỀ THI THỬ ĐẠI HỌC
Câu 2 Tính các tính phân sau đây:
1 R
π
3
−
π
3
xsin x
0
x2ex
x2+ 4x + 4dx
3 Re 2
1
ln3x+ 2 ln x
x1 +p2 ln2x+ 1
1
1 − x5
x(1 + x5)dx
5 R
π
3
π
4
π 2
0
sin x + cos x
3 + sin 2x dx
7 R5
2
ln(√
x− 1 + 1)
x− 1 +√x− 1dx 8 Rln 5
ln 2
dx (10e− x
− 1)√ex
− 1
9 Rln 8
ln 3
e2xdx
ex
− 3√ex
0
xln(x + 2)
√
4 − x2 dx
11 R
π
2
0
sin xdx (sin x +√
π 2
0
1 + sin x
1 + cos x
exdx
13 R√3
1
dx
0
xex
(x + 1)2dx
15 Re
1
ln x − 2
π 2
0
sin xdx
√
1 + cos2x
17 R1
0
x2ex
+
4
√x
1 +√x
π 2 π 4
x− π4(1 − sin 2x)dx
1 + sin 2x
19 R
π
2
0
sin 2x − 3 cos x
0
√
1 − sin xdx
21 R
π
2
0
sin 2xdx
3 + 4 sin x − cos 2x 22 R63
0
dx
√x + 1 +√3x
+ 1
23 R2√2
√
3
xln x
√
1 + x2
π 3 π 4
3
p sin3x− sin x sin3x dx
25 R
π
4
0
sin 4x
2 + sin x − cos xdx 26 R1
0
dx
e2x+ ex
27 R
π
6
π
8
cot x − tan x − 2 tan 2x
sin 4x dx 28 Rln 2
0
(2ex
+ 3)dx
ex
+ 2e− x
+ 3
29 R
π
6
0
3 sin2x− sin x cos x sin x − cos x dx 30 R
π 3
0
xex
[4 + 4√
2 sin(x + π
4) + sin 2x]dx (1 + cos x)2
31 R
π
6
0
sin 3xdx
1
√ 3
0
x8dx (x4
− 1)