AN AN ACT ACT ACT OF OF OF KINDNESS KINDNESS KINDNESS IS IS IS NEVER NEVER NEVER WASTED WASTED WASTED !!!!.
Trang 1‹ CÁC 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
3.TÍNH :
4/I = 3 2
4
3tg x dx
π
π
∫
6
(2cotg x 5)dx
π
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 3
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
17/I = 2 sin x2
4
e sin 2x dx
π
π
18/ I = ∫4 +
0 2
2
cos
π
x
etgx
19/ I = ∫2
4
4
sin 1
π
π x dx
20/ I = ∫4
0 6
cos 1 π
x dx
21/I =2cos2x(si n4x cos4x)dx
0
+
∫
π
22/ I = 2 3
0
cos xdx
π
23/ I =
3 2
0
4sin x
dx
1 cosx
π
+
24/ I =
1
0
x 1 x dx−
∫
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)+
∫
Trang 232/I =
7
3
3
0
x 1
dx 3x 1
+
+
∫
33/I =
2
3 2 0
(x 3) x− −6x 8 dx+
∫
34/I =
1
3
1
dx
x 4 x−
∫
35/I =
4
2 2
1
dx
x 16 x−
∫
36*/I =
6
2
2 3
1
dx
x x −9
∫
37/I =
2
1
x 4 x dx
−
−
∫
38/I =
2
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
−
−
+
∫
ln 3 1
dx
∫
47/I =4 2
6
1
dx sin x cot gx
π
π
∫
48/I =
e 1
ln x 2 ln x
dx x
+
∫
49/I =e
1
sin(ln x)
dx x
∫
0
x (x −1) dx
∫
0
(1 2x)(1 3x 3x ) dx + + +
∫ 52/I =
2
3 1
1 dx
x 1 x+
∫
6
tg x cot g x 2dx
π
π
∫
0
(1 x ) dx−
∫
55*/I =1 2x
0
1 dx
e +3
∫
56/I =
x
ln 3
x 3 0
e
dx (e +1)
∫
1
x(e x 1)dx
−
∫
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
x 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
+
∫
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
+ +
∫
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
x 1
+ +
∫
2 sin x
π
Trang 376/I =e
1
cos(ln x)dx
π
∫
0
4 x dx+
∫
78/I =2
1
x
dx
1+ x 1−
∫
79/I =
e
1
1 3ln x ln x
dx x
+
∫
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
∫
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
∫
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
∫
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
−
−
∫
1
x 2x x 2 dx
−
∫
98/I =
3 4
4
cos 2x 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
−
∫
1
−
∫
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
∫
0
x.sin x cos xdx
π
∫
110*/I =
2 x 1
2 0
x e
dx (x 2)+
∫
0
e sin xdx
π
∫
112/I =
2 2 1
1
x ln(1 )dx
x +
∫
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
∫
Trang 4118/I =4
0
1
dx cos x
π
∫
0
1 dx cos x
π
∫
120/I =1 3 x2
0
x e dx
∫
0
e sin x cos xdx
π
∫
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)+
∫
2
sin 2x
dx (2 sin x)
−π∫ +
129/I =1
2 0
x 3
dx
−
∫
130/I =1 3
0
4x
dx (x +1)
∫
131/I =
1
0
1
dx (x +4x +3)
∫
3
3 sin x
π
133/I =
3 3
6
4sin x
dx
1 cos x
π
π −
∫
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
0
sin x
dx (tg x 1) cos x
π
+
∫
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 sin x
dx
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−
∫
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
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
∫
0
x cos xdx
π
∫
159/I =
1 0
cos x dx
∫
Trang 5160/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
∫
0
x cos x sin x dx
π
∫
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
∫
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
∫
1 e
ln x
dx (x 1)+
∫
178/I =
1 2 0
1 x
x ln dx
1 x
+
−
∫
179/I =2
3
cos x.ln(1 cos x)dx
π
π
−
∫
180/2 sin x2 3
0
e sin x cos x dx
π
∫
0
sin 2x
dx
1 sin x
π
+
∫
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
+ +
∫
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 =
2 4
0
1 2sin x
dx
1 sin 2x
π
− +
∫
195/I =
3 2 0
x 2x
dx
x 1
+ +
∫
196/I =3
2 4
tgx
dx cos x 1 cos x
π
197/I =
2
2 1
x 1 ( ) dx
x 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−
∫
Trang 6202/I =
2
2 1
ln(1 x)
dx x
+
∫
203/I =2
0
sin 2x
dx
1 cos x
π
+
∫
0
dx
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
∫
0
cos x.cos 4x 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−
∫
0
x
dx
4 x−
∫
214/I =
1
4 2
2
0
x
dx
x −1
∫
215/I =2 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
∫
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
cos x
dx cos x 1
π
+
∫
226/I =
7 3 3 0
x 1
dx 3x 1
+ +
∫
227/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∫x (1 x) dx2 − 3
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
−
− +
∫
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)+
∫
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
∫
0
x sin x cos xdx
π
∫
2 cosx cosx cos xdx π
π
−
∫
240*/I =
1
2 1
ln( x a x)dx
−
+ +
∫
0
1 sin x
dx (1 cos x)e
π
− +
∫
242/I =2
0
sin 2x sin x
dx cos3x 1
π
+ +
∫
dx
π
∫
Trang 7244/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
∫
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
+ +
∫
2
cosx cosx cos xdx
π
π
−
∫
256/I =3 4
4
tg xdx
π
π
∫
0
1 sin x
e dx
1 cos x
π
+ +
∫
0
(1 x ) dx−
∫
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 )
− +
∫
0
cos x
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 cos 2x
π
+
∫
266/I =2
3
1
dx sin x 1 cos x
π
∫
0
sin x
dx cos x 3
π
+
∫
268/I =
2
0
sin x
dx x
π
∫
0
sinxcosx(1 cosx) dx
π
+
∫
270/I =
4 0
sin x cos x
dx sin x cos x 1
π
−
∫
271/I =
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
x 2x 10x 1
dx
x 2x 9
+ +
∫
275/I =
3 1
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
+ +
∫
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
+ + +
∫
1
(x 1) ln x dx−
∫
278/I =
1
3 0
x dx (2x 1)+
∫
279/I =7
2
1
dx
2 x 1+ +
∫
Trang 8283/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
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 cos 2x
π
+
∫
289/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
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
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
1 x 1 x
−∫ + + +
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
∫
3
cos x
dx (1 cos x)
π
π −
∫
307/I =4 3
0
tg x dx
π
∫
308*/I=
2 x
sin x
dx
3 1
π
+
∫
309*/I =
1
2x 1
1 dx
3 e
−∫ +
310*/I =2
0
sin x
dx cos x sin x
π
+
∫
311/I =
4 2
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
+
∫
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
0
cos x
dx cos 3cos x 3
π
∫
318*/Tìm x> 0 để x 2 t 2
0
t e
dt 1 (t 2)+ =
∫
319*/I =3
2 4
tan x
dx cos x cos x 1
π
∫
0
3x 6x 1dx
∫
321*/I =4 5
0
tg x dx
π
∫
Trang 9322/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
+
∫
0
sin x
dx cos x 1
π
+
∫
326/I =3
2 6
cos 2x
dx
1 cos 2x
π
π −
∫
0
t g x 1
tg x 1
π
− +
∫
328*/I =1
3 1
2
x dx
x + 1
∫
4 1
dx x
−
∫
0
e
d x
∫
331/I =
1
4
e
2 1
e
1
dx
x cos (ln x 1)
π
+
∫
332/I = 3
1
1
dx
x (1 x ) +
∫
333*/I =4
0
ln(1 tgx)dx
π
+
∫
334,Cmr
2
2 0
dx
π
+
7
−
≤∫ + + − ≤
336,Cmr 3
1
1
dx
−
≤∫ ≤
337,Cmr
1 2
0
dt t
2
4 2
0
2
2
x x
e dx e e
−
≤∫ ≤
339, Cmr
1
<
+
340,
2
1
x
=
∫
341,
2 3
2 5
dx I
=
+
∫
342,
1
0
I=∫x 1 x dx−
343,
2
0
sin 2x sin x
1 3cos x
π
+
=
+
∫
344,
1
2 0
I=∫ 1 x dx− 345,
1 2 0
dx I
1 x
= +
∫
346,
1 2 0
dx I
=
∫
347,
e
1
1 3ln x ln x
x
+
=∫
348,
ln 5
ln3
dx I
=
∫
0
sin 2x cos x
1 cos x
π
= +
∫
3 2 2
I=∫ln x −x dx
2 sin x 0
π
1
2x 0
I=∫ x 2 e dx− 353,
2 2 0
I=∫ x −x dx
0
x
1 cos2x
π
= +
∫
3 x 0
I=∫x e dx 356,
2
0
I=π∫ x sin xdx
0
1 2sin x
1 s in2x
π
−
= +
∫
358,
3 3 1
dx I
x x
= +
∫
3
π
π
2 2 1
5 x 1
−
=
− −
∫
1
2 0
1 x
=
+
∫
2
e
e
ln x ln ln x
x
+
= ∫
6
cos x
sin x
π
π
= ∫
364,
1 2 6 0
x dx I
=
−
∫
3 5 4
4
dx I
sin x cos x
π
π
=∫
0
sin 2x
sin x 6 sin x 5
π
=
∫
0
I=∫πsin xe dx
0
π
= ∫
4
sin x cos x
π
π
+
=
+
∫
Trang 10
AN
AN ACT ACT ACT OF OF OF KINDNESS KINDNESS KINDNESS IS IS IS NEVER NEVER NEVER WASTED WASTED WASTED !!!!