333 BÀI TOÁN TÍCH PHÂN LUYỆN THI ĐẠI HỌC
1/ Cho hàm số : f(x)= x.sinx+x2 Tìm nguyên hàm của hàm số g(x)= x.cosx
biết rằng nguyên hàm này triệt tiêu khi x=kπ
2/Định m để hàm số: F(x) = mx3+(3m+2)x2 -4x+3 là một nguyên hàm của hàm số:
f(x) = 3x2 +10x-4
3/Tìm họ nguyên hàm của hàm số: f(x)= cos3x.sin8x
TÍNH :
4/I =
3
2
4
3tg x dx
π
π
∫
5/I = 4 2
6
(2cotg x 5)dx
π
6/I = 2
0
1 cos x
dx
1 cos x
π
−
+
∫
7/ I =∫2
0
π
sin2 x.cos2xdx
8/I = ∫3
0
π
(2cos2 x-3sin2 x)dx
9 / I =
2
2
4
π
−π
π−
π+
10 / I = ∫
−
3
6
π
π
(tgx-cotgx)2 dx
11/ I = 4 4
0
cos x dx
π
12 / I = 2 3
0
sin x dx
π
∫
13*/ I =
2
3
sin x sin x
cot gx dx sin x
π π
−
∫
14/I = 2 4
0
sin x dx
π
∫
15/I = ∫3
4
2 2
2
cos 2 sin 1
π
16/I = ∫4
6
π
π
cotg2x dx
17/I = 2 sin x2
4
e sin 2x dx
π
18/ I = ∫4 +
0 2
2 cos
π
x
etgx
.
34/I =
1
3
1
dx
x 4 x−
∫
Trang 219/ I = ∫2
4
4
sin
1
π
20/ I = ∫4
0 6
cos 1
π
x dx
21/I = 2cos2x(si n4x cos4 x)dx
0
+
∫
π
22/ I = 2 3
0
cos xdx
π
∫
23/ I = 2 3
0
4sin x
dx
1 cosx
π
+
24/ I =
1
0
x 1 x dx−
∫
25/I =
1
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
−
∫
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
0
(x 3) x− −6x 8 dx+
∫
.
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
0
x (x +4) dx
∫
39/I =
2 4
4 3 3
dx x
−
∫
40*/I =
2 2
2 2
dx
−
−
+ +
∫
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
−
∫
46/I =
ln 3
x 0
1 dx
e +1
∫
47/I =
4 2 6
1
dx sin x cot gx
π
π∫
48/I =
e 1
ln x 2 ln x
dx x
+
∫
.
Trang 349/I =
e
1
sin(ln x)
dx x
∫
50/I =
1
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
6
tg x cot g x 2dx
π
∫
54/I =
1
2 3 0
(1 x ) dx−
∫
55*/I =
1
2x
0
1 dx
e +3
∫
56/I =
x
ln 3
0
e
dx (e +1)
∫
57/I =
0
1
−
∫
58/I =26 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 cos 2x
π
+
∫
61/I =
2x
ln 5
x
ln 2
e
dx
e −1
∫
62/I =
2
e
1
.ln xdx x
+
∫
63/I =
2 1
0
x
dx (x 1) x 1+ +
∫
64/I =2
0
sin x.sin 2x.sin 3xdx
π
∫
0
cos 2x(sin x cos x)dx
π
+
∫
66*/I =2 3 3
0
( cos x sin x )dx
π
−
∫
67/I =
7 3
2
x
dx
1 x+ −2x
∫
68*/I =2
0
4cos x 3sin x 1
dx 4sin x 3cos x 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
0
x 1 x dx+
∫
74**/I =
1 2 0
ln(1 x)
dx
+ +
∫
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−
∫
.
Trang 479/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
cos x
dx
6 5sin x sin x
π
∫
95*/I =
2
e 2 e
ln x
ln x −
∫
96/I =
3 2 4
x 4 dx
−
−
∫
97/I =
2
1
−
∫
98/I =
3 4
4
cos 2x 1dx
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 =
2 1
−
∫
104*/I = 2
0
x sin x dx
1 cos x
π
+
∫
105*/I =
1
1
1
dx (x 1)(4 1)
106*/I =
4 1
x 1
x dx
1 2
107/I =
2
4 0
x sin xdx
π
∫
108/I =
2
4 0
x cos xdx
π
∫
Trang 5109/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
1 x
+
−
∫
115/I =
2 t
1
ln x
x
∫
116/I =3
0
sin x.ln(cos x)dx
π
∫
117/I =e2 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
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
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)
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
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
∫
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
∫
.
Trang 6137/I = 4 3
0
sin x
dx (tg x 1) cos x
π
+
∫
138/I =
3
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 3cos x
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
144/I =3 3
0
sin x
dx cos x
π
∫
145/I =
1
0
x 1 xdx−
∫
146/I =
6
4
x 2 x 2
−
∫
147/I =
0
2 1
1
dx
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
−
−
∫
151/I =
1
x 0
1 dx
3 e+
∫
152/I =
1
2
2x 0
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 = 2 4
0
cos x
dx cos x sin x
π
+
∫
156/I =
1 0
3
dx
x 9+ − x
∫
157/I =
0
x sin 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
x sin x dx
π
∫
162/I =
2
4 0
x cos x dx
π
∫
163/I = 2
0
x cos x sin x dx
π
∫
164/I = 6 2
0
x cos x sin x dx
π
∫
165/I =
4 x 1
e dx
∫
166/I =4 3x
0
e sin 4x dx
π
∫
Trang 7167/I = 2x 2
0
e sin x dx
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
x ln x dx
∫
171/I =
1
e
2
1
ln x dx
∫
172/I =
e
1
x(2 ln x)dx−
∫
173/I =
2
e
2 e
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
1 x
+
−
∫
179/I =
2
3
cos x.ln(1 cos x)dx
π
180/2 2
0
e sin x cos 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
dx
x 3
+
∫
185/I =
4 2 1
1 dx
x (x 1)+
∫
186/I =
1 2 0
ln(1 x)
dx
+ +
∫
187/I
4 1
6 0
1 x
dx
1 x
+ +
∫
188/I =
1
0
x 1 x dx+
∫
189/I =
x 1
0
e
dx
e +e−
∫
190/I=
e
1 e
ln x dx
∫
191/I =2 sin x
0
(e cos x)cos x dx
π
+
∫
192/I =2
0
sin 2x.cos x
dx
1 cos x
π
+
∫
193/I =2
0
sin 2x sin x
dx
1 3cos x
π
+ +
∫
194/I =4 2
0
1 2sin x
dx
1 sin 2x
π
− +
∫
195/I =
3 2 0
dx
+ +
∫
196/I =3
2 4
tgx
dx cos x 1 cos x
π
∫
Trang 8197/I =
2
2 1
x 1
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 =2 2008
0
dx
π
+
∫
205/I =2
0
sin x.ln(1 cos x)dx
π
+
∫
206/I =
2 3
2 1
dx x
+
∫
207/I =4 3
2 0
sin x
dx cos x
π
∫
208/I =2 2
0
cos x.cos 4x dx
π
∫
209/I =
1
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
sin 3x
dx cos x 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
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
dx
x 1
+ +
∫
224/I =
1
0
(1 x) e dx+
∫
225/I =2
2 0
cos x
dx cos x 1
π
+
∫
226/I =
7 3 3 0
x 1
dx 3x 1
+ +
∫
.
Trang 9227/I =2
6
1 sin 2x cos 2x
dx cos x sin x
π
π
+
∫
228/I =
x 2 1
2x 0
(1 e )
dx
1 e
+
+
∫
229/I =
3
0
x (1 x) dx−
∫
230/I = 2 3
2 0
sin x.cos x
dx cos x 1
π
+
∫
231/I =
1
2
2
0
4x 1
dx
−
∫
232*/I = 2
0
x sin x.cos xdx
π
∫
233/I =2
0
cos x
dx cos 2x 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
x sin x cos xdx
π
∫
2
cos x 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
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
0
x (1 x ) dx−
∫
250/I =2
0
sin x
dx
1 sin x
π
+
∫
251/I =2
0
cos x
dx
7 cos 2x
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
cos x sin x
dx
3 sin 2x
π π
+ +
∫
Trang 10
255/I = 2 3
2
cos x 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
cos x
dx
1 sin x
π
−
∫
264/I =3 2
6 0
sin x
dx cos x
π
∫
265/I =6 3
0
sin x sin x
dx cos 2x
π
+
∫
265/I =2
3
1
dx sin x 1 cos x
π
266/I =
3
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
∫
0
sin x cos x(1 cos x) dx
π
+
∫
270/I =4 4 4
0
sin x cos x
dx sin x cos x 1
π
−
∫
271/I =4 4 4
0
sin x cos x
dx sin x cos x 1
π
−
∫
272/I =2
0
sin x cos x cos x
dx sin x 2
π
+ +
∫
273/I =
1
1 x 3 a
e dx x
∫
274/I =
1
2 0
dx
∫
275/I =
3 1
0
x
dx (x +1)
∫
276/I =
1 3 0
3 dx
x +1
∫
277*/I =
4 1 6 0
dx
+ +
∫
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−
∫
.
Trang 11281*/I =
2 1
2 0
x ln(x 1 x )
dx
1 x
+
∫
282/I =
4
2 1
(x 1) ln x dx−
∫
283/I =
3
2
0
x ln(x 1)dx+
∫
284/I =
3 2
2
1
3x
dx
x +2x 1+
∫
285/I =
1
0
4x 1
dx
−
∫
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 cos 2x
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 x sin xdx
π
∫
292/I =2 4 4
0
cos 2x(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
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
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 =2 3
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
307/I =4 3
0
tg x dx
π
∫
Trang 12308*/I =
1
2x 1
1 dx
3 e
309*/I =
2 x
sin x
dx
π
310*/I =2
0
sin x
dx cos x sin x
π
+
∫
311/I =2 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
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 =2 3
0
cos x
dx cos 3cos x 3
π
∫
318*/Tìm x> 0 sao cho
2 t x
2 0
t e
dt 1
+
∫
319*/I =
3
2 4
tan x
dx cos x cos x 1
π
∫
320*/I =
1
2 0
∫
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 =2 5
0
sin x
dx cos x 1
π
+
∫
326/I =
3
2 6
cos 2x
dx
1 cos 2x
π
∫
327*/I =4 2
0
t gx 1
tgx 1
π
− +
∫
328*/I =
1 3 1 2
x dx
x +1
∫
329*/I =
2 4 1
x x
dx x
−
∫
330/I =
x
ln 3
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
+
∫
.