333 bài tích phân luyện thi ĐH

333 BÀI TOÁN TÍCH PHÂN LUYỆN THI ĐẠI HỌC



1/ Cho hàm số : f(x)= x.sinx+x
2
. Tìm nguyên hàm của hàm số g(x)= x.cosx
biết rằng nguyên hàm này triệt tiêu khi x=k
π

2/Định m để hàm số: F(x) = mx
3
+(3m+2)x
2
-4x+3 là một nguyên hàm của hàm số:
f(x) = 3x
2
+10x-4.
3/Tìm họ nguyên hàm của hàm số: f(x)= cos
3
x.sin8x.

TÍNH :

4/I =
3
2
4
3tg x dx
π
π

∫

5/I =
4
2
6
(2cotg x 5)dx
π
π
+
∫

6/I =
2
0
1 cosx
dx
1 cos x
π
−
+
∫

7/ I =
∫
2
0
π
sin
2

x.cos
2
xdx
8/I =
∫
3
0
π
(2cos
2
x-3sin
2
x)dx
9 / I =
2
2
s i n ( x )
4
d x
s i n ( x )
4
π
− π
π
−
π
+
∫

10 / I =

∫
−
3
6
π
π
(tgx-cotgx)
2
dx

11/ I =
4
4
0
cos xdx
π
∫



12 / I =
2
3
0
sin x dx
π
∫

13*/ I =
3

3
2
3
sin x sin x
cotgx dx
sin x
π
π
−
∫

14/I =
2
4
0
sin x dx
π
∫

15/I =
∫
3
4
22
2
cos
2
sin
1
π

π
xx
dx

16/I =
∫
4
6
π
π
cotg2x dx

17/I =
2
2
sin x
4
e sin 2x dx
π
π
∫

18/ I =
∫
+
4
0
2
2
cos

π
x
e
tgx



.


34/I =
1
2 2
3
1
dx
x 4 x
−
∫

19/ I =
∫
2
4
4
sin
1
π
π
x

dx
20/ I =
∫
4
0
6
cos
1
π
x
dx

21/I =
dxxxnsix )cos(2cos
44
2
0
+
∫
π

22/ I =
2
3
0
cos xdx
π
∫

23/ I =

3
2
0
4sin x
dx
1 cosx
π
+
∫

24/ I =
1
3 2
0
x 1 x dx
−
∫

25/I =
1
5 2
0
x 1 x dx
+
∫

26/I =
1
0
x

dx
2x 1
+
∫

27/I =
1
x
0
1
dx
e 4
+
∫

28/I =
2
x
1
1
dx
1 e
−
−
∫

29/I =
2x
2
x

0
e
dx
e 1
+
∫

30/I =
x
1
x
0
e
dx
e 1
−
−
+
∫

31/I =
e
2
1
ln x
dx
x(ln x 1)
+
∫

32/I =
7
3
3
0
x 1
dx
3x 1
+
+
∫

33/I =
2
3
2
0
(x 3) x 6x 8dx
− − +
∫

.

35/I =
4
2
2
1
dx
x 16 x

−
∫

36*/I =
6
2
2 3
1
dx
x x 9
−
∫

37/I =
2
2 2
1
x 4 x dx
−
−
∫

38/I =
2
2 3
0
x (x 4) dx
+
∫

39/I =
2
4
4 3
3
x 4
dx
x
−
∫

40*/I =
2
2
2
2
x 1
dx
x x 1
−
−
+
+
∫

41/I =
ln 2
x
0
e 1dx

−
∫

42/I =
1
0
1
dx
3 2x
−
∫

43/I =
2
5
0
sin xdx
π
∫

44*/I =
3
0
1
dx
cos x
π
∫

45/I =

2x
1
x
0
e
dx
e 1
−
−
+
∫

46/I =
ln3
x
0
1
dx
e 1
+
∫

47/I =
4
2
6
1
dx
sin x cot gx
π

π
∫

48/I =
3
2
e
1
ln x 2 ln x
dx
x
+
∫

.


49/I =
e
1
sin(ln x)
dx
x
∫

50/I =
1
3 4 5
0
x (x 1) dx

−
∫

51/I =
1
2 3
0
(1 2x)(1 3x 3x ) dx
+ + +
∫

52/I =
2
3
1
1
dx
x 1 x
+
∫

53/I =
3
2 2
6
tg x cot g x 2dx
π
π
+ −
∫

54/I =
1
2 3
0
(1 x ) dx
−
∫

55*/I =
1
2x
0
1
dx
e 3
+
∫

56/I =
x
ln3
x 3
0
e
dx
(e 1)+
∫

57/I =

0
2x
3
1
x(e x 1)dx
−
+ +
∫

58/I =
2
6
3 5
0
1 cos x sin x.cos xdx
π
−
∫

59*/I =
2 3
2
5
1
dx
x x 4
+
∫

60/I =

4
0
x
dx
1 cos2x
π
+
∫

61/I =
2x
ln5
x
ln 2
e
dx
e 1
−
∫

62/I =
2
e
1
x 1
.ln xdx
x
+
∫

63/I =
2
1
0
x
dx
(x 1) x 1
+ +
∫


64/I =
2
0
sin x.sin 2x.sin3xdx
π
∫

65/I =
2
4 4
0
cos2x(sin x cos x)dx
π
+
∫

66*/I =
2
3 3

0
( cosx sin x)dx
π
−
∫

67/I =
7
3
8 4
2
x
dx
1 x 2x+ −
∫

68*/I =
2
0
4cos x 3sin x 1
dx
4sin x 3cosx 5
π
− +
+ +
∫

69/I =
9
3

1
x. 1 xdx
−
∫

70/I =
2
3
0
x 1
dx
3x 2
+
+
∫

71*/I =
6
0
x
sin dx
2
π
∫

72*/I =
2
0
x
dx

2 x 2 x
+ + −
∫

73/I =
3
3 2
0
x . 1 x dx
+
∫

74**/I =
1
2
0
ln(1 x)
dx
x 1
+
+
∫

75/I =
2
0
sin x
dx
sin x cos x
π

+
∫

76/I =
e
1
cos(ln x)dx
π
∫

77*/I =
2
2
0
4 x dx
+
∫

78/I =
2
1
x
dx
1 x 1
+ −
∫


.

79/I =
e
1
1 3ln x ln x
dx
x
+
∫

80/I =
3
2
2
ln(x x)dx
−
∫


81/I =
e
2
1
(ln x) dx
∫

82/I =
2
e
e
ln x

dx
x
∫

83/I =
2
e
1
ln x
dx
ln x
∫

84/I =
2
2
1
x ln(x 1)dx
+
∫

85/I =
3
2
3
1
dx
x 3
+
∫

86/I =
1
2
0
1
dx
4 x
−
∫

87/I =
2
4
0
sin xdx
π
∫

88/I =
3
2
6
ln(sin x)
dx
cos x
π
π
∫

89/I =
2
1
cos(ln x)dx
∫

90*/I =
2
2
0
ln( 1 x x)dx
+ −
∫

91*/I =
3
2
2
1
dx
x 1
−
∫

92/I =
3
8
1
x 1
dx

x
+
∫

93/I =
3
3
2
1
x
dx
x 16
−
∫


.

94/I =
6
2
0
cosx
dx
6 5sin x sin x
π
− +
∫

95*/I =

2
e
2
e
1 1
( )dx
ln x
ln x
−
∫

96/I =
3
2
4
x 4 dx
−
−
∫

97/I =
2
3 2
1
x 2x x 2 dx
−
− − +
∫

98/I =

3
4
4
cos2x 1dx
π
π
+
∫

99/I =
0
cos x sin xdx
π
∫

100/I =
2
0
1 sin xdx
π
+
∫

101/I =
3
4
4
sin 2x dx
π
π

∫

102/I =
0
1 sin xdx
π
−
∫

103/I =
1
3
2
1
ln(x x 1) dx
−
 
+ +
 
 
∫

104*/I =
2
0
xsin x
dx
1 cos x
π
+

∫

105*/I =
1
2 x
1
1
dx
(x 1)(4 1)
−
+ +
∫

106*/I =
4
1
x
1
x
dx
1 2
−
+
∫

107/I =
2
4
0
xsin xdx

π
∫

108/I =
2
4
0
x cos xdx
π
∫


109/I =
6
2
0
x.sin x cos xdx
π
∫

110*/I =
2 x
1
2
0
x e
dx
(x 2)
+
∫

111/I =
2x 2
0
e sin xdx
π
∫


112/I =
2
2
1
1
x ln(1 )dx
x
+
∫


113/I =
e
2
1
e
ln x
dx
(x 1)
+
∫

114/I =
1
2
0
1 x
x.ln dx
1 x
+
−
∫

115/I =
2
t
1
ln x
dx I 2
x
 
⇒
<
 
 
∫

116/I =
3
0
sin x.ln(cos x)dx

π
∫

117/I =
2
e
2
1
cos (ln x)dx
π
∫

118/I =
4
0
1
dx
cos x
π
∫

119*/I =
4
3
0
1
dx
cos x
π
∫

120/I =
2
1
3 x
0
x e dx
∫

121/I =
2
2
sin x 3
0
e .sin x cos xdx
π
∫

122/I =
2
4
0
sin 2x
dx
1 cos x
π
+
∫

123/I =

1
2
0
3
dx
x 4x 5
− −
∫

124/I =
2
2
1
5
dx
x 6x 9
− +
∫

125/I =
1
2
5
1
dx
2x 8x 26
−
+ +
∫

126/I =
1
0
2x 9
dx
x 3
+
+
∫

127/I =
4
2
1
1
dx
x (x 1)
+
∫


128*/I =
0
2
2
sin 2x
dx
(2 sin x)
−π
+

∫

129/I =
1
2
0
x 3
dx
(x 1)(x 3x 2)
−
+ + +
∫

130/I =
1
3
0
4x
dx
(x 1)
+
∫

131/I =
1
4 2
0
1
dx
(x 4x 3)

+ +
∫

132/I =
3
3
2
0
sin x
dx
(sin x 3)
π
+
∫

133/I =
3
3
6
4sin x
dx
1 cos x
π
π
−
∫

134/I =
3
2

6
1
dx
cos x.sin x
π
π
∫

135/I =
3
0
sin x.tgxdx
π
∫

136/I =
3
4
1
dx
sin 2x
π
π
∫


.
137/I =
3
4

2 2 5
0
sin x
dx
(tg x 1) .cos x
π
+
∫

138/I =
3
2 2
3
1
dx
sin x 9cos x
π
π
−
+
∫

139/I =
2
2
cos x 1
dx
cos x 2
π
π

−
−
+
∫

140/I =
2
0
1 sin x
dx
1 3cosx
π
+
+
∫

141/I =
2
0
cos x
dx
sin x cos x 1
π
+ +
∫

142/I =
4
2
1

1
dx
x (x 1)
+
∫

143/I =
1
3
3
1
dx
x 4 (x 4)
−
+ + +
∫

144/I =
3
3
0
sin x
dx
cos x
π
∫

145/I =
1
0

x 1 xdx
−
∫

146/I =
6
4
x 4 1
. dx
x 2 x 2
−
+ +
∫

147/I =
0
2
1
1
dx
x 2x 9
−
+ +
∫

148/I =
3
2
1
1

dx
4x x
−
∫

149/I =
2
2
1
4x x 5 dx
−
− +
∫

150/I =
2
2
2
2x 5
dx
x 4x 13
−
−
+ +
∫

151/I =
1
x
0

1
dx
3 e
+
∫

152/I =
1
4x 2x
2
2x
0
3e e
dx
1 e
+
+
∫

153/I =
4
2
7
1
dx
x 9 x
+
∫

154/I =

2
x 2
0
e sin xdx
π
∫

155/I =
4
2
4 4
0
cos x
dx
cos x sin x
π
+
∫


156/I =
1
0
3
dx
x 9 x
+ −
∫

157/I =

0
xsin xdx
π
∫

158/I =
2 2
0
x cos xdx
π
∫

159/I =
1
0
cos x dx
∫

160/I =
1
0
sin x dx
∫

161/I =
2
4
0
xsin x dx
π

∫

162/I =
2
4
0
x cos x dx
π
∫

163/I =
2
0
x cos xsin xdx
π
∫

164/I =
6
2
0
xcos xsin x dx
π
∫

165/I =
4
x
1
e dx

∫

166/I =
4
3x
0
e sin 4xdx
π
∫

167/I =
2x 2
0
e sin xdx
π
∫

168/I =
2 x
1
2
0
x e
dx
(x 2)
+
∫

169/I =
e

1
(1 x)ln x dx
+
∫

170/I =
e
2
1
xln xdx
∫

171/I =
1
e
2
1
ln xdx
∫

172/I =
e
1
x(2 ln x)dx
−
∫

173/I =
2
e

2
e
1 1
( )dx
ln x
ln x
−
∫

174/I =
2
2
1
(x x)ln x dx
+
∫

175/I =
2
2
1
1
x ln(1 )dx
x
+
∫

176/I =
2
5

1
ln x
dx
x
∫

177/I =
e
2
1
e
ln x
dx
(x 1)
+
∫

178/I =
1
2
0
1 x
x ln dx
1 x
+
−
∫

179/I =
2

3
cosx.ln(1 cosx)dx
π
π
−
∫

180/
2
2
sin x 3
0
e sin xcos x dx
π
∫

181/I=
2
4
0
sin 2x
dx
1 sin x
π
+
∫


.
182/I =

2
4
0
sin 2x
dx
1 cos x
π
+
∫

183/I =
2
2
1
5
dx
x 6x 9
− +
∫

184/I =
2
1
0
x 3x 2
dx
x 3
+ +
+
∫

185/I =
4
2
1
1
dx
x (x 1)
+
∫

186/I =
1
2
0
ln(1 x)
dx
x 1
+
+
∫

187/I
4
1
6
0
1 x
dx
1 x

+
+
∫

188/I =
1
15 8
0
x 1 x dx
+
∫

189/I =
x
1
x x
0
e
dx
e e
−
+
∫

190/I=
e
1
e
ln x dx
∫

191/I =
2
sin x
0
(e cosx)cosxdx
π
+
∫

192/I =
2
0
sin 2x.cosx
dx
1 cos x
π
+
∫

193/I =
2
0
sin 2x sin x
dx
1 3cos x
π
+
+
∫

194/I =
2
4
0
1 2sin x
dx
1 sin 2x
π
−
+
∫

195/I =
5 3
3
2
0
x 2x
dx
x 1
+
+
∫

196/I =
3
2
4
tgx

dx
cosx 1 cos x
π
π
+
∫

197/I =
2
2
1
x 1
( ) dx
x 2
−
−
+
∫

198/I =
4
2
0
x.tg x dx
π
∫

199/I =
5
3

( x 2 x 2 )dx
−
+ − −
∫

200/I =
4
1
2
dx
x 5 4
−
+ +
∫

201/I =
2
1
x
dx
x 2 2 x
+ + −
∫


202/I =
2
2
1
ln(1 x)

dx
x
+
∫

203/I =
2
0
sin 2x
dx
1 cos x
π
+
∫

204/I =
2008
2
2008 2008
0
sin x
dx
sin x cos x
π
+
∫

205/I =
2
0

sin x.ln(1 cos x)dx
π
+
∫

206/I =
2
3
2
1
x 1
dx
x
+
∫

207/I =
3
4
2
0
sin x
dx
cos x
π
∫

208/I =
2
2

0
cos x.cos4x dx
π
∫

209/I =
1
2x x
0
1
dx
e e
+
∫

210/I =
e
2
1
e
ln x
dx
(x 1)
+
∫

211/I =
1
0
1

dx
x 1 x
+ +
∫

212/I =
2
1
2
0
x
dx
4 x
−
∫

213/I =
1
2
0
x
dx
4 x
−
∫

214/I =
1
4
2

2
0
x
dx
x 1
−
∫

215/I =
2
0
sin3x
dx
cosx 1
π
+
∫

216/I =
2
2
2
2
0
x
dx
1 x
−
∫

217/I =
2
2
4
1
1 x
dx
1 x
−
+
∫


218/I =
3
7
3
2
0
x
dx
1 x
+
∫

219/I =
x
ln 2
x
0

1 e
dx
1 e
−
+
∫

220/I =
1
0
x 1 x dx
−
∫

221/I =
1
2
0
x 1dx
+
∫

222/I =
2
3 3
0
(cos x sin x)dx
π
+
∫

223/I =
2
3
0
x 1
dx
x 1
+
+
∫

224/I =
1
2 2x
0
(1 x) .e dx
+
∫

225/I =
2
2
0
cosx
dx
cos x 1
π
+
∫

226/I =
7
3
3
0
x 1
dx
3x 1
+
+
∫

.
227/I =
2
6
1 sin 2x cos2x
dx
cosx sin x
π
π
+ +
+
∫

228/I =
x 2
1
2x

0
(1 e )
dx
1 e
+
+
∫

229/I =
3
2 3
0
x (1 x) dx
−
∫

230/I =
3
2
2
0
sin x.cos x
dx
cos x 1
π
+
∫

231/I =
1

2
2
0
4x 1
dx
x 3x 2
−
− +
∫

232*/I =
2
0
xsin x.cos xdx
π
∫

233/I =
2
0
cosx
dx
cos2x 7
π
+
∫

234/I =
4
2

1
1
dx
x (x 1)
+
∫

235/I =
2
2 3
0
sin 2x(1 sin x) dx
π
+
∫

236/I =
2
3
0
x 1
dx
3x 2
+
+
∫

237/I =
4
2

7
1
dx
x x 9
+
∫

238/I =
3 4
0
xsin x cos xdx
π
∫

239/I =
2
3
2
cosx cos x cos xdx
π
π
−
−
∫

240*/I =
1
2
1
ln( x a x)dx

−
+ +
∫

241/I =
2
x
0
1 sin x
dx
(1 cos x)e
π
−
+
∫


242/I =
2
0
sin 2x sin x
dx
cos3x 1
π
+
+
∫

243/I =
4

2 2
0
sin 2x
dx
sin x 2cos x
π
+
∫

244/I =
2
3
2
2
0
x
dx
1 x
−
∫

245/I =
2
3
2
2
0
x
dx
1 x

−
∫

246/I =
2
1
2
2
2
1 x
dx
x
−
∫

247/I =
2
1
2
0
x
dx
4 x
−
∫

248/I =
2
2
2

3
1
dx
x x 1
−
∫

249/I =
1
5 3 6
0
x (1 x ) dx
−
∫

250/I =
2
0
sin x
dx
1 sin x
π
+
∫

251/I =
2
0
cosx
dx

7 cos2x
π
+
∫

252/I =
4
2
1
1
dx
(1 x)x+
∫

253/I =
2
3
0
x 1
dx
3x 2
+
+
∫

254*/I =
3
4
cosx sin x
dx

3 sin 2x
π
π
+
+
∫


.


255/I =
2
3
2
cosx cos x cos xdx
π
π
−
−
∫

256/I =
3
4
4
tg xdx
π
π
∫

257*/I =
2
x
0
1 sin x
e dx
1 cos x
π
+
+
∫

258/I =
1
2 3
0
(1 x ) dx
−
∫

259/I =
4
2
0
x.tg xdx
π
∫

260/I=

2
2 2
0
1
dx
(4 x )+
∫

261/I =
2
1
3
0
3x
dx
x 2
+
∫

262*/I =
5
2
5
1
1 x
dx
x(1 x )
−
+
∫

263/I =
3
2
0
cosx
dx
1 sin x
π
−
∫

264/I =
2
3
6
0
sin x
dx
cos x
π
∫

265/I =
3
6
0
sin x sin x
dx
cos2x

π
+
∫

265/I =
2
3
1
dx
sin x 1 cosx
π
π
+
∫

266/I =
3
6 2
1
1
dx
x (1 x )
+
∫


.

267/I =
2

2
0
sin x
dx
cos x 3
π
+
∫

268/I =
2
0
sin x
dx
x
π
∫

269/I =
2
2
0
sin x cosx(1 cos x) dx
π
+
∫

270/I =
4 4
4

0
sin x cos x
dx
sin x cosx 1
π
−
+ +
∫


271/I =
4 4
4
0
sin x cos x
dx
sin x cosx 1
π
−
+ +
∫

272/I =
2
0
sin x cos x cosx
dx
sin x 2
π
+

+
∫

273/I =
1
1
x
3
a
e
dx
x
∫

274/I =
3 2
1
2
0
x 2x 10x 1
dx
x 2x 9
+ + +
+ +
∫

275/I =
3
1
2 3

0
x
dx
(x 1)+
∫

276/I =
1
3
0
3
dx
x 1
+
∫

277*/I =
4
1
6
0
x 1
dx
x 1
+
+
∫

278/I =
1

3
0
x
dx
(2x 1)+
∫

279/I =
7
2
1
dx
2 x 1
+ +
∫

280/I =
3
2
2
1
2
1
dx
x 1 x−
∫

.

281*/I =

2
1
2
0
x ln(x 1 x )
dx
1 x
+ +
+
∫

282/I =
4
2
1
(x 1) ln xdx
−
∫

283/I =
3
2
0
x ln(x 1)dx
+
∫

284/I =
3
2

2
1
3x
dx
x 2x 1
+ +
∫

285/I =
1
3 2
0
4x 1
dx
x 2x x 2
−
+ + +
∫

286/I =
1
2
2
1
2
1
dx
(3 2x) 5 12x 4x
−
+ + +

∫

287/I =
1
0
1
dx
x 1 x
+ +
∫

288/I =
2
0
cos x
dx
2 cos2x
π
+
∫

289/I =
2
4
cos x sin x
dx
3 sin 2x
π
π
+

+
∫

290/I =
2
3 3
0
(cos x sin x)dx
π
+
∫

291/I =
2
5 4
0
cos xsin xdx
π
∫

292/I =
2
4 4
0
cos2x(sin x cos x)dx
π
+
∫

293/I =

2
0
1
dx
2 sin x
π
+
∫

294/I =
2
0
1
dx
2 cos x
π
−
∫


295/I =
2
2
2
3
1
dx
x x 1
−
∫

296/I =
3
7
3
2
0
x
dx
1 x
+
∫

297*/I =
2
3
1
1
dx
x 1 x
+
∫

298/I =
3
1
2
0
x
dx

x 1 x
+ +
∫

299/I =
1
2
1
1
dx
1 x 1 x
−
+ + +
∫



300/I =
3
4
6
1
dx
sin x cos x
π
π
∫

301/I =
2

0
cos x
dx
cos x 1
π
+
∫

302/I =
2
0
cos x
dx
2 cos x
π
−
∫

303/I =
2
0
sin x
dx
sin x 2
π
+
∫

304/I =
3

2
0
cos x
dx
cos x 1
π
+
∫

305/I =
2
0
1
dx
2cos x sin x 3
π
+ +
∫


306/I =
2
2
3
cos x
dx
(1 cos x)
π
π
−

∫

307/I =
4
3
0
tg x dx
π
∫



308*/I =
1
2x
1
1
dx
3 e
−
+
∫

309*/I =
2
x
sin x
dx
3 1
π

− π
+
∫

310*/I =
2
0
sin x
dx
cos x sin x
π
+
∫

311/I =
4
2
4 4
0
sin x
dx
cos x sin x
π
+
∫

312*/I =
2
2
0

tgx
dx
1 ln (cos x)
π
−
∫

313*/I =
2
0
sin x
dx
cos x sin x
π
+
∫

314*/I =
1
x 2
1
1
dx
(e 1)(x 1)
−
+ +
∫

315*/I =
1

3x 1
0
e dx
+
∫

316*/I =
2
1
2
0
x
dx
x 4
+
∫

317*/I =
3
2
4 2
0
cos x
dx
cos 3cos x 3
π
− +
∫

318*/Tìm x> 0 sao cho

2 t
x
2
0
t e
dt 1
(t 2)
=
+
∫

319*/I =
3
2
4
tan x
dx
cos x cos x 1
π
π
+
∫

320*/I =
1
2
0
3x 6x 1dx
− + +
∫

321*/I =
4
5
0
tg x dx
π
∫

322/I =
4
3
6
cotg x dx
π
π
∫

323/I =
3
4
4
tg x dx
π
π
∫

324*/I =
4

0
1
dx
2 tgx
π
+
∫

325/I =
5
2
0
sin x
dx
cos x 1
π
+
∫

326/I =
3
2
6
cos2x
dx
1 cos 2x
π
π
−
∫

327*/I =
4
2
0
t gx 1
( ) dx
tgx 1
π
−
+
∫

328*/I =
1
3
1
2
x
dx
x 1
+
∫

329*/I =
3
3
2
4
1

x x
dx
x
−
∫

330/I =
x
ln3
x x
0
e
dx
(e 1) e 1
+ −
∫

331/I =
1
4
e
2
1
e
1
dx
x cos (ln x 1)
π
−
+

∫

333*/I =
4
0
ln(1 tgx)dx
π
+
∫