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 =
4
2 6
(2cotg x 5)dx
π
5/ I =∫2
0
π
sin 2 x.cos 2 xdx
6/I = ∫3
0
π
(2cos 2 x-3sin 2 x)dx
7/ I =
2
2
4
π
−π
π−
π+
8 / I = ∫
−
3
6
π
π
(tgx-cotgx) 2 dx
9/I = 2cos2x(si n4x cos4 x)dx
0
+
∫
π
10/ I = 2 3
0
4sin x
dx
1 cosx
π
+
11/ I =
1
0
x 1 x dx−
∫
12/I =
1
0
x 1 x dx+
∫
13/I =
1
0
x
dx 2x 1+
∫
14/I =
3 2 4
3tg x dx
π π
15/ I =2 3
0
sin x dx
π
16/I = 2
0
1 cos x
dx
1 cos x
π
− +
17/I = 2 4
0
sin x dx
π
∫
18/I = ∫4
6
π
π
cotg2x dx
19/ I = ∫4 +
0 2
2
cos
π
x
etgx
20/ I = 4 4
0
cos x dx
π
∫
21/ I = ∫2
4
4
sin 1
π
π x dx
22/ I =∫4
0 6
cos 1
π
x
dx
23*/ I =
2
3
sin x sin x
cot gx dx sin x
π π
−
∫
24/I = ∫3
4
2 2
2
cos 2 sin 1
π
25/I = 2
2 sin x 4
e sin 2x dx
π π
26/I =
1
3
1
dx
x 4 x−
∫
27/I =
4
2 2
1
dx
x 16 x−
∫
28*/I =
6
2
2 3
1
dx
x x −9
∫
29/I =
2
1
x 4 x dx
−
−
∫
30/I =
2
0
x (x +4) dx
∫
31/I =
2 4
4 3 3
dx x
−
∫
32*/I =
2 2
2 2
dx
−
−
+ +
∫
33/I =
ln 2
x 0
e −1dx
∫
Trang 234/I =
x
1
x
0
e
dx
−
∫
35/I =
e
2
1
ln x
dx x(ln x 1)+
∫
36/I =
7
3
3
0
x 1
dx 3x 1
+
+
∫
37/I =
2
0
(x 3) x− −6x 8 dx+
∫
38/I =
e
1
sin(ln x)
dx x
∫
39/I =
1
0
x (x −1) dx
∫
40/I =
1
2 3 0
(1 2x)(1 3x 3x ) dx+ + +
41/I =
3
6
tg x cot g x 2dx
π
∫
42/I =
x
ln 3
0
e
dx (e +1)
∫
43/I =
0
1
−
∫
0
1 cos x sin x.cos xdx
π
−
∫
45*/I =
2 3
2 5
1 dx
x x +4
∫
46/I =4
0
x
dx
1 cos 2x
π
+
∫
47/I =
2x
ln 5
x
ln 2
e
dx
e −1
∫
48/I =
2 1
0
x
dx (x 1) x 1+ +
∫
49/ I = 2 3
0
cos xdx
π
∫
50/I =
1 x 0
1 dx
e +4
∫
51/I =
2
x 1
1 dx
1 e− −
∫
52/I =
2x 2 x 0
e dx
e +1
∫
53*/I =3
0
1 dx cos x
π
∫
54/I =
2x 1 x 0
e
dx
−
∫
55/I =
ln 3
x 0
1 dx
e +1
∫
56/I =
2
3 1
1 dx
x 1 x+
∫
57/I =
1
2 3 0
(1 x ) dx−
∫
58*/I =
1 2x 0
1 dx
e +3
∫
59/I =
9 3 1
x 1 xdx−
∫
60/I =
2 3 0
x 1
dx 3x 2
+ +
∫
61*/I = 6
0
x sin dx 2
π
∫
62**/I =
1 2 0
ln(1 x)
dx
+ +
∫
63/I =
2 e 1
.ln xdx x
+
∫
64*/I =
2
2 0
4 x dx+
∫
65/I =
e
2 1
(ln x) dx
∫
66/I =
1 0
1 dx
3 2x−
∫
67/I =2 5 0
sin xdx
π
∫
68/I =
4 2 6
1
dx sin x cot gx
π π
∫
69/I =
e 1
ln x 2 ln x
dx x
+
∫
70/I =2
0
sin x.sin 2x.sin 3xdx
π
∫
0
cos 2x(sin x cos x)dx
π
+
∫
0
( cos x sin x )dx
π
−
∫
73/I =
7 3
2
x
dx
1 x+ −2x
∫
74*/I =2
0
4cos x 3sin x 1
dx 4sin x 3cos x 5
π
∫
75*/I =
2 0
x
dx
2 x+ + 2 x−
∫
76/I =
3
0
x 1 x dx+
∫
77/I =2
0
sin x
dx sin x cos x
π
+
∫
78/I =
e 1
cos(ln x)dx
π
∫
79/I =
2 1
x
dx
1+ x 1−
∫
Trang 380/I =
e
1
1 3ln x ln x
dx x
+
∫
81/I =
3
2
2
ln(x −x)dx
∫
82/I =
2
2 1
x ln(x +1)dx
∫
83/I =
1
2 0
1
dx
4 x−
∫
84/I =
3
2 6
ln(sin x)
dx cos x
π
π
∫
85*/I =
2
2 0
ln( 1 x+ −x)dx
∫
86*/I =
3
2 2
1 dx
x −1
∫
0
x.sin x cos xdx
π
∫
0
e sin xdx
π
∫
89/I =
2
2
1
1
x ln(1 )dx
x
+
∫
90/I =
e
2 1
e
ln x
dx (x 1)+
∫
91/I =
1
2
0
1 x
1 x
+
−
∫
92/I =
2 t
1
ln x
x
∫
93/I =3
0
sin x.ln(cos x)dx
π
∫
94/I =e2 2
1
cos (ln x)dx
π
∫
95/I =
2
e e
ln x dx x
∫
96/I =
2
e 1
ln x dx
ln x
∫
97/I =
3 2 3
1 dx
x +3
∫
98/I =
2
4 0
sin xdx
π
∫
99/I =
2 1
cos(ln x)dx
∫
100/I =
3 8 1
x 1 dx x
+
∫
101/I =
3 3 2 1
x dx
x −16
∫
102/I =
3 4
4
sin 2x dx
π π
∫
103*/I =
2 x 1
2 0
x e
dx (x 2)+
∫
104*/I =
4 1
x 1
x dx
1 2
105/I =
2
4 0
x sin xdx
π
∫
106/I =
2
4 0
x cos xdx
π
∫
107/I =
1 0
2x 9
dx
x 3
+ +
∫
108/I =
4 2 1
1 dx
x (x 1)+
∫
109/I =
1 3 0
4x dx (x +1)
∫
110/I =6
2 0
cos x
dx
6 5sin x sin x
π
∫
111*/I =
2
e 2 e
ln x
ln x −
∫
112/I =
3 2 4
x 4 dx
−
−
∫
113/I =
2
1
−
∫
114/I =
3 4
4
cos 2x 1dx
π
115/I =
0
cos x sin xdx
π
∫
116/I =
2 0
1 sin xdx
π +
∫
117/I =
0
1 sin xdx
π
−
∫
118/I =
2 1
−
∫
0
x sin x
dx
1 cos x
π +
∫
120*/I =
1
1
1
dx (x 1)(4 1)
121/I =
1 2 0
3
dx
x −4x 5−
∫
122/I =
2 2 1
5
dx
x −6x 9+
∫
123/I =
1 2 5
1
dx
124*/I =
0
2 2
sin 2x
dx (2 sin x)
125/I =
1
2 0
x 3
dx (x 1)(x 3x 2)
−
∫
Trang 4126/I =2 sin x2 3
0
e sin x cos xdx
π
∫
0
sin x
dx (tg x 1) cos x
π
+
∫
128/I =
3
3
1
dx sin x 9cos x
π
π
129/I =
2
2
cos x 1
dx cos x 2
π
π
−
− +
∫
130/I = 2
0
1 sin x
dx
1 3cos x
π
+
+
∫
131/I =2
0
cos x
dx sin x cos x 1
π
∫
132/I =
1
3 3
1
dx
133/I =
6
4
x 2 x 2
−
∫
134/I =
0
2 1
1
dx
135/I =
2
2 1
4x x 5 dx
−
∫
136/I =
2
2 2
2x 5
dx
−
−
∫
137/I = 2x 2
0
e sin x dx
π
∫
138/I =
2 x
1
2 0
x e
dx (x 2)+
∫
139/I =
e
1
(1 x)ln x dx+
∫
140/I =
e
1
x(2 ln x)dx−
∫
141/I =4
0
1 dx cos x
π
∫
142*/I =4
3 0
1 dx cos x
π
∫
1
3 x 0
x e dx
∫
144/I =2
4 0
sin 2x
dx
1 cos x
π +
∫
145/I =
3
4
1 dx sin 2x
π π
∫
146/I =
4 2 1
1 dx
x (x 1)+
∫
147/I =3 3
0
sin x
dx cos x
π
∫
148/I =
1 0
x 1 xdx−
∫
149/I =
3
2 1
1
dx 4x x−
∫
150/I =
1
x 0
1 dx
3 e+
∫
151/I =
0
x sin xdx
π
∫
152/I =
1 0
cos x dx
∫
153/I =
1 0
sin x dx
∫
154/I =
e 2 1
x ln x dx
∫
155/I =
1 e 2 1
ln x dx
∫
156/I =
1
0
1
dx (x +4x +3)
∫
2 0
sin x
dx (sin x 3)
π
+
∫
158/I =
3 3
6
4sin x
dx
1 cos x
π
∫
159/I =
3
2 6
1
dx cos x.sin x
π π
∫
160/I =3
0
sin x.tgxdx
π
∫
161/I =
1
2
2x 0
dx
1 e
+ +
∫
162/I =
4
2 7
1
dx
x 9 x+
∫
163/I = 2 x 2
0
e sin xdx
π
∫
0
cos x
dx cos x sin x
π
+
∫
165/I =
1 0
3
dx
x 9+ − x
∫
0
x cos xdx
π
∫
167/I =
2
4 0
x sin x dx
π
∫
168/I =
2
4 0
x cos x dx
π
∫
0
x cos x sin x dx
π
∫
Trang 5170/I =
2
e
2 e
ln x
ln x −
∫
171/I =
2
2
1
(x +x)ln x dx
∫
172/I =
2
2
1
1
x ln(1 )dx
x
+
∫
173/I =
2
3
cos x.ln(1 cos x)dx
π
∫
0
e sin x cos x dx
π
∫
175/I= 2
4 0
sin 2x
dx
1 sin x
π
+
∫
176/I =
5
3
( x 2 x 2 )dx
−
+ − −
∫
177/I =
4
1
2
dx
x 5 4
178/I =
2
1
x
dx
x 2+ + 2 x−
∫
0
dx
π
+
∫
180/I =2
0
sin x.ln(1 cos x)dx
π
+
∫
181/I =2 2
0
cos x.cos 4x dx
π
∫
182/I =
1
0
1
dx
e +e
∫
183/I =
1
0
1
dx
x 1+ + x
∫
184/I =
2
6
1 sin 2x cos 2x
dx cos x sin x
π
π
+
∫
185/I =
4 x 1
e dx
∫
186/I =
2 5 1
ln x dx x
∫
187/I =
e
2 1
e
ln x
dx (x 1)+
∫
188/I =
1 2 0
1 x
1 x
+
−
∫
189/I =
4 2 1
1 dx
x (x 1)+
∫
190/I
4 1
6 0
1 x
dx
1 x
+ +
∫
191/I=
e
1 e
ln x dx
∫
192/I =
2
2 1
x 1
x 2
−
− +
∫
193/I =4 2
0
x.tg x dx
π
∫
194/I =
2 2 1
ln(1 x)
dx x
+
∫
195/I =2
0
sin 2x
dx
1 cos x
π +
∫
196/I =
2 3 2 1
dx x
+
∫
197/I =4 3
2 0
sin x
dx cos x
π
∫
198/I =
e
2 1
e
ln x
dx (x 1)+
∫
199/I =
2 1
2 0
x dx
4 x−
∫
200/I = 6 2
0
x cos x sin x dx
π
∫
201/I =4 3x
0
e sin 4x dx
π
∫
202/I =2
4 0
sin 2x
dx
1 cos x
π +
∫
203/I =
2 2 1
5
dx
x −6x 9+
∫
204/I =
2 1 0
dx
x 3
+
∫
205/I =
1 2 0
ln(1 x)
dx
+ +
∫
206/I =
1
0
x 1 x dx+
∫
207/I =
x 1
0
e
dx
e +e−
∫
208/I =2 sin x
0
(e cos x)cos x dx
π
+
∫
209/I =2
0
sin 2x.cos x
dx
1 cos x
π +
∫
210/I =2
0
sin 2x sin x
dx
1 3cos x
π
+ +
∫
0
1 2sin x
dx
1 sin 2x
π
− +
∫
212/I =
3 2 0
dx
+ +
∫
213/I =
3
2 4
tgx
dx cos x 1 cos x
π
∫
Trang 6214/I =
x 2 1
2x 0
(1 e )
dx
1 e
+
+
∫
215/I =
3
0
x (1 x) dx−
∫
2 0
sin x.cos x
dx cos x 1
π
+
∫
217/I =
1
2
2
0
4x 1
dx
−
∫
0
x sin x.cos xdx
π
∫
219/I =2
0
cos x
dx cos 2x 7
π
+
∫
220/I =
4
2
1
1 dx
x (x 1)+
∫
0
sin 2x(1 sin x) dx
π
+
∫
222/I =
4
2 7
1
dx
x x +9
∫
0
x sin x cos xdx
π
∫
224/I =
2
3 2
cos x cos x cos xdx
π
π
−
−
∫
225*/I =
1
2 1
ln( x a x)dx
−
+ +
∫
226/I =2
x 0
1 sin x
dx (1 cos x)e
π
− +
∫
227/I =
2
3 2
cos x cos x cos xdx
π
π
−
−
∫
228/I =
1
2 0
x dx
4 x−
∫
229/I =
1 4 2 2 0
x dx
x −1
∫
230/I =
2 2 2
2 0
x dx
1 x−
∫
231/I =
2 2
4 1
1 x
dx
1 x
− +
∫
232/I =
3 7
0
x dx
1 x+
∫
233/I =
x
ln 2
x 0
1 e
dx
1 e
− +
∫
234/I =
1 0
x 1 x dx−
∫
235/I =
1 2 0
x +1dx
∫
236/I =
2 3 0
x 1
dx 3x 2
+ +
∫
237/I =
2 3 2
2 0
x dx
1 x−
∫
238/I =
2 3 2
2 0
x dx
1 x−
∫
239/I =
2 1
2 2 2
1 x
dx x
−
∫
240/I =
2 1
2 0
x
dx
4 x−
∫
241/I =
4
2 1
1 dx (1 x)x+
∫
242/I =
2
0
sin x
dx x
π
∫
243/I =2
0
sin 3x
dx cos x 1
π
+
∫
0
(cos x sin x)dx
π
+
∫
245/I =
2 3 0
dx
x 1
+ +
∫
246/I =
1
2 2x 0
(1 x) e dx+
∫
247/I =2
2 0
cos x
dx cos x 1
π
+
∫
248/I =
7 3 3 0
x 1
dx 3x 1
+ +
∫
249/I =2
0
sin 2x sin x
dx cos3x 1
π
+ +
∫
250/I =4
0
sin 2x
dx sin x 2cos x
π
+
∫
251/I =
2
2 2
3
1 dx
x x −1
∫
252/I =
1
0
x (1 x ) dx−
∫
253/I =2
0
sin x
dx
1 sin x
π +
∫
254/I =2
0
cos x
dx
7 cos 2x
π +
∫
255/I =
2 3 0
x 1
dx 3x 2
+ +
∫
256*/I =
3
4
cos x sin x
dx
3 sin 2x
π π
+ +
∫
Trang 7257*/I =2 x
0
1 sin x
e dx
1 cos x
π
+
+
∫
258/I =
1
2 3 0
(1 x ) dx−
∫
259*/I =
5 2
5 1
1 x
dx x(1 x )
− +
∫
260/I =3
2 0
cos x
dx
1 sin x
π
−
∫
0
sin x sin x
dx cos 2x
π
+
∫
262/I =
2
3
1
dx sin x 1 cos x
π
∫
263/I =
3
1
1
dx
x (1 x )+
∫
264*/I =
2 1
2 0
x ln(x 1 x )
dx
1 x
+
∫
265/I =
4
2 1
(x 1) ln x dx−
∫
266/I =
3
2
0
x ln(x 1)dx+
∫
267/I =
3 2
2
1
3x
dx
x +2x 1+
∫
268/I =
1
0
4x 1
dx
−
∫
269/I =
1
2
2 1
2
1
dx (3 2x) 5 12x 4x
270/I =
1
0
1
dx
x + 1 x+
∫
271/I =2
0
cos x
dx
2 cos 2x
π
+
∫
272/I =
3 4 4
tg xdx
π
π∫
273/I =4 2
0
x.tg xdx
π
∫
274/I=
2
2 2 0
1 dx (4 x )+
∫
275/I =
2 1
3 0
3x
dx
x +2
∫
276/I =3 2
6 0
sin x
dx cos x
π
∫
277/I =
1
1 x 3 a
e dx x
∫
278/I =
3 1
0
x
dx (x +1)
∫
279/I =
1 3 0
3 dx
x +1
∫
280*/I =
4 1 6 0
dx
+ +
∫
281/I =
1
3 0
x dx (2x 1)+
∫
282/I =
3 7
0
x dx
1 x+
∫
283/I =2
0
cos x
dx cos x 1
π
+
∫
284/I =2
0
cos x
dx
2 cos x
π
−
∫
285/I =2
0
sin x
dx sin x 2
π
+
∫
286/I =2 3
0
cos x
dx cos x 1
π
+
∫
287/I =2
2 0
sin x
dx cos x 3
π
+
∫
0
sin x cos x(1 cos x) dx
π
+
∫
0
sin x cos x
dx sin x cos x 1
π
−
∫
0
sin x cos x
dx sin x cos x 1
π
−
∫
291/I =2
0
sin x cos x cos x
dx sin x 2
π
+ +
∫
292/I =
1
2 0
dx
∫
293/I =
7 2
1
dx
2 x 1+ +
∫
294/I =
3 2
2 1
2
1 dx
x 1 x−
∫
295/I =
2
2 2
3
1 dx
x x −1
∫
296*/I =
2
3 1
1 dx
x 1 x+
∫
297/I =
3 1
2 0
x
dx
x+ 1 x+
∫
298/I =
1
2 1
1
dx
299/I =
3 4 6
1
dx sin x cos x
π π
∫
300/I = 2
0
1
dx 2cos x sin x 3
π
∫
Trang 8301/I =
2
4
cos x sin x
dx
3 sin 2x
π
π
+ +
∫
0
(cos x sin x)dx
π
+
∫
0
cos x sin xdx
π
∫
0
cos 2x(sin x cos x)dx
π
+
∫
305/I =2
0
1 dx
2 sin x
π
+
∫
306*/I =2
0
sin x
dx cos x sin x
π
+
∫
0
sin x
dx cos x sin x
π
+
∫
308*/I =2
2 0
tgx
dx
1 ln (cos x)
π
−
∫
309*/I =2
0
sin x
dx cos x sin x
π
+
∫
310*/I =
1
1
1
dx (e 1)(x 1)
0
cos x
dx cos 3cos x 3
π
∫
312*/Tìm x> 0 sao cho
2 t x
2 0
t e
dt 1
+
∫
313/I =4 3
0
tg x dx
π
∫
314*/I =4 5
0
tg x dx
π
∫
315/I =
4 3 6
cotg x dx
π
π∫
316/I =
3 4 4
tg x dx
π π
∫
317*/I = 4
0
1 dx
2 tgx
π +
∫
318*/I =
1
2x 1
1 dx
3 e
319*/I =
2 x
sin x
dx
π
320/I =2
0
1 dx
2 cos x
π
−
∫
321*/I =
1 3x 1 0
e + dx
∫
322*/I =
2 1 2 0
x
dx
x +4
∫
323/I =
2
2 3
cos x
dx (1 cos x)
π
∫
324/I =2 5
0
sin x
dx cos x 1
π
+
∫
325/I =
3
2 6
cos 2x
dx
1 cos 2x
π
∫
0
t gx 1
tgx 1
π
− +
∫
327*/I =
1 3 1 2
x dx
x +1
∫
328*/I =
2 4 1
x x
dx x
−
∫
329/I =
x
ln 3
0
e
dx (e +1) e −1
∫
330/I =
1 4
e
2 1
e
1
dx
x cos (ln x 1)
π−
+
∫
331*/I =4
0
ln(1 tgx)dx
π +
∫
332*/I =
3
2 4
tan x
dx cos x cos x 1
π
∫
333*/I =
1
2 0
∫
.