✶✳✶✳ ❚Ý❝❤ ♣❤➞♥ ❦❤➠♥❣ ①➳❝ ➤Þ♥❤
✶✳ ❑✐Ó♠ tr❛ ❧➵✐ ❝➳❝ ➤➻♥❣ t❤ø❝ s❛✉ ➤➞②
✭❛✮
Z√x2±adx =x
2√x2±a±a
2ln x+√x2±a+C
tr♦♥❣ ➤ã
a > 0
✳
✭❜✮
Z√a2−x2dx =x
2√a2−x2+a2
2arcsin x
a+C
tr♦♥❣ ➤ã
a > 0
✳
✭❝✮
Zdx
√a2−x2= arcsin x
a+C
tr♦♥❣ ➤ã
a > 0
✳
✭❞✮
Zdx
√x2±a= ln x+√x2±a+C
tr♦♥❣ ➤ã
a > 0
✳
✭❡✮
Zdx
x2+a2=1
aarctan x
a+C
tr♦♥❣ ➤ã
a6= 0
✳
✭❢✮
Zdx
a2−x2=1
2aln
a+x
a−x
+C
tr♦♥❣ ➤ã
a6= 0
✳
✷✳ ❑✐Ó♠ tr❛ ❧➵✐ ❝➳❝ ➤➻♥❣ t❤ø❝ s❛✉ ➤➞②
✭❛✮
Zeax cos bx =bsin bx +acos bx
a2+b2eax +C
tr♦♥❣ ➤ã
a6= 0
✳
✭❜✮
Zeax sin bx =asin bx −bcos bx
a2+b2eax +C
tr♦♥❣ ➤ã
a6= 0
✳
✭❝✮
Zdx
sin x= ln tan x
2+C
tr♦♥❣ ➤ã
x6=kπ
✳
✭❞✮
Zdx
cos x= ln tan x
2+π
4+C
tr♦♥❣ ➤ã
x6=π
2+kπ
✳
✸✳ ❚Ý♥❤ ❝➳❝ tÝ❝❤ ♣❤➞♥ s❛✉✿
•Zdx
x4−1
✳
•Z1 + 2x2
x2(1 + x2)dx
✳
•Z√x2+ 1 + √1−x2
√1−x4dx
✳
•Z√x2+ 1 −√1−x2
√x4−1dx
✳
•Z√x4+x−4+ 2
x3dx
✳
•Z23x−1
ex−1dx
✳
•Z22x−1
√2xdx
✳
•Zdx
x(2 +
❧♥
2x)
✳
•Z3
√
❧♥
2x
xdx
✳
•Zex+e2x
1−exdx
✳
•Zexdx
1 + ex
✳
•Zsin2x
2dx
✳
Phép tính tích phân trên R
1
Bài tập Giải tích II
K53 -Toán tin
ĐHKHTN
•Z
❝♦t❣
2xdx
✳
•Z√1 + sin 2xdx
✱
x∈0,π
2
✳
•Zecos xsin xdx
✳
•Zexcos exdx
✳
•Z1
1 + cos xdx
✳
•Zdx
sin x+ cos x
✳
•Z1 + cos x
(x+ sin x)3dx
✳
•Zsin 2x
p1−4 sin2x
dx
✳
•Zsin x
p2−sin2x
dx
✳
•Zsin xcos x
p3−sin4x
dx
✳
•Z
❛r❝❝♦t❣
3x
1 + 9x2dx
✳
•Zx+√
❛r❝t❣
2x
1 + 4x2dx
✳
•Z
❛r❝
sin x−
❛r❝
cos x
√1−x2dx
✳
•Zx+
❛r❝
sin32x
√1−4x2dx
✳
•Zx+
❛r❝
cos3/2x
√1−x2dx
✳
•Zx|x|dx
✳
•Z(2x−3)|x−2|dx
✳
•Zf(x)dx
✱
f(x) = (1−x2,|x|61,
1− |x|,|x|>1.
✹✳ ❚Ý♥❤ ❝➳❝ tÝ❝❤ ♣❤➞♥ s❛✉
✭❛✮
Ze2x
4
√ex+ 1dx
✳
✭❜✮
Zdx
√ex+ 1
✳
✭❝✮
Ze2x
ex−1dx
✳
✭❞✮
Z√1 +
❧♥
x
xdx
✳
✭❡✮
Z√1 +
❧♥
x
x
❧♥
xdx
✳
✭❢✮
Zdx
ex/2+ex
✳
✭❣✮
Z
❛r❝t❣
√x
√x
dx
1 + x
✳
✭❤✮
Z√e3x+e2xdx
✳
✭✐✮
Ze2x2+2x−1(2x+ 1)dx
✳
✭❥✮
Zdx
√ex−1
✳
✭❦✮
Ze2xdx
√e4x+ 1
✳
✭❧✮
Z2xdx
√1−4x
✳
✭♠✮
Zdx
1 + √x+ 1
✳
✭♥✮
Zx+ 1
x√x−2dx
✳
✭♦✮
Zdx
√ax +b+m
✳
✭♣✮
Zdx
3
√x(3
√x−1)
✳
✭q✮
Zdx
(1 −x2)3/2
✳
✭r✮
Zdx
(x2+a2)3/2
✳
✭s✮
Zdx
(x2−1)3/2
✳
✭t✮
Z√a2−x2dx
✳
✭✉✮
Z√a2+x2dx
✳
✭✈✮
Zx2
√a2+x2dx
✳
✭✇✮
Zdx
x2√x2+a2
✳
✭①✮
Zx2dx
√a2−x2
✳
✭②✮
Zdx
x√x2−a2
✳
✺✳ ❚Ý♥❤ ❝➳❝ tÝ❝❤ ♣❤➞♥ s❛✉✿
Phép tính tích phân trên R
2
Bài tập Giải tích II
K53 -Toán tin
ĐHKHTN
•Zx2xdx
✳
•Zx2e−xdx
✳
•Zx3e−x2dx
✳
•Z(x3+x)e5xdx
✳
•Z
❛r❝
sin xdx
✳
•Zx
❛r❝
sin xdx
✳
•Zx2
❛r❝
sin 2xdx
✳
•Z
❛r❝t❣
xdx
✳
•Z
❛r❝t❣
√xdx
✳
•Zx3
❛r❝t❣
xdx
✳
•Z(
❛r❝t❣
x)2xdx
✳
•Z(
❛r❝
sin x)2dx
✳
•Z
❛r❝
sin x
√x+ 1 dx
✳
•Z
❛r❝
sin x
x2dx
✳
•Zx
❛r❝t❣
x
√1 + x2dx
✳
•Z
❛r❝
sin √x
√1−xdx
✳
•Zln xdx
✳
•Z√xln2xdx
✳
•Zln(x+√16 + x2)dsx
✳
•Zxln(x+√1 + x2)
√1 + x2dx
✳
•Zsin xln(
t❣
x)dx
✳
•Zx2ln(1 + x)dx
✳
•Zx2sin 2xdx
✳
•Zx3cos(2x2)dx
✳
•Zexsin xdx
✳
•Z3xcos xdx
✳
•Re3x(sin 2x−cos 2x)dx
✳
•Zxe2xsin 5xdx
✳
•Zx2exsin xdx
✳
•Zx2excos xdx
✳
•Zx2sin(ln x)dx
✳
✻✳ ❚×♠ ❝➠♥❣ t❤ø❝ tr✉② ❤å✐ ❝❤♦ ❝➳❝ tÝ❝❤ ♣❤➞♥ s❛✉✿
•In=Zxneaxdx
✱
a6= 0
✳
•In=Zlnnxdx
✳
•In=Zxαlnnxdx
✱
α6=−1
✳
•In=Zxndx
√x2+a
✱
n > 2
✳
•In=Zsinnxdx
✱
n > 2
✳
•In=Zcosnxdx
✱
n > 2
✳
•In=Zdx
cosnx
✱
n > 2
✳
✼✳ ❚Ý♥❤ ❝➳❝ tÝ❝❤ ♣❤➞♥ s❛✉
✭❛✮
Zxdx
(x+ 1)(x+ 2)(x−3)
✳
✭❜✮
Z2x4+ 5x2−2
2x3−x−1dx
✳
✭❝✮
Z2x3+x2+ 5x+ 1
(x2+ 3)(x2−x+ 1)dx
✳
✭❞✮
Zx4+x2+ 1
x(x−2)(x+ 2)dx
✳
Phép tính tích phân trên R
3
Bài tập Giải tích II
K53 -Toán tin
ĐHKHTN
✭❡✮
Zdx
x(x−1)(x2−x+ 1)2
✳
✭❢✮
Zx4−x2+ 1
(x2−1)(x2+ 4)(x2−2)dx
✳
✭❣✮
Z3x2+ 5x+ 12
(x2+ 3)(x2+ 1)dx
✳
✭❤✮
Z(x4+ 1)dx
x5+x4−x3−x2
✳
✭✐✮
Zx3+x+ 1
x4−1dx
✳
✭❥✮
Zx4
1−x4dx
✳
✭❦✮
Z3x+ 5
(x2+ 2x+ 2)2dx
✳
✭❧✮
Zx4−2x2+ 2
(x2−2x+ 2)2dx
✳
✭♠✮
Zx2+ 2x+ 7
(x−2)(x2+ 1)3dx
✳
✭♥✮
Zx2
(x+ 2)2(x+ 1)dx
✳
✭♦✮
Zx2+ 1
(x−1)3(x+ 3)dx
✳
✭♣✮
Zdx
x5−x2
✭q✮
Z3x2+ 8
x3+ 4x2+ 4xdx
✳
✭r✮
Z2x5+ 6x3+ 1
x4+ 3x2dx
✳
✭s✮
Zx3+ 4x2−2x+ 1
x4+xdx
✳
✭t✮
Zx3−3
x4+ 10x2+ 25dx
✳
✽✳ ❚Ý♥❤ ❝➳❝ tÝ❝❤ ♣❤➞♥
✭❛✮
Zdx
√2x−1−3
√2x−1
✳
✭❜✮
Zxdx
(3x−1)√3x−1
✳
✭❝✮
Zr1−x
1 + x
dx
x
✳
✭❞✮
Z3
rx+ 1
x−1
dx
x+ 1
✳
✭❡✮
Z√x+ 1 −√x−1
√x+ 1 + √x−1dx
✳
✭❢✮
Zxdx
√x+ 1 −3
√x+ 1
✳
✭❣✮
Z(x−2)r1 + x
1−xdx
✳
✭❤✮
Z3
rx+ 1
x−1
dx
(x−1)3
✳
✭✐✮
Zdx
p(x−1)3(x−2)
✳
✭❥✮
Zdx
3
p(x−1)2(x+ 1)
✳
✭❦✮
Zdx
3
p(x+ 1)2(x−1)4
✳
✭❧✮
Zdx
4
p(x−1)3(x+ 2)5
✳
✭♠✮
Zdx
3
p(x−1)7(x+ 1)2
✳
✭♥✮
Zdx
6
p(x−7)7(x−5)5
✳
✭♦✮
Zdx
n
p(x−a)n+1(x−b)n−1
✱
a6=b
✳
✭♣✮
Z√x+ 1 −√x−1
√x+ 1 + √x−1dx
✳
Phép tính tích phân trên R
4
Bài tập Giải tích II
K53 -Toán tin
ĐHKHTN
✾✳ ❙ö ❞ô♥❣ ❝➳❝ ♣❤Ð♣ t❤Õ ❊✉❧❡r tÝ♥❤ ❝➳❝ tÝ❝❤ ♣❤➞♥ s❛✉
✭❛✮
Zdx
x√x2+x+ 1
✳
✭❜✮
Zdx
(x−2)√−x2+ 4x−3
✳
✭❝✮
Zdx
(x+ 1)√1 + x−x2
✳
✭❞✮
Zdx
(x−1)√x2+x+ 1
✳
✭❡✮
Z(x−1)dx
(x2+ 2x)√x2+ 2x
✳
✭❢✮
Z5x+ 4
√x2+ 2x+ 5dx
✳
✶✵✳ ❚Ý♥❤ ❝➳❝ tÝ❝❤ ♣❤➞♥ ✈✐ ♣❤➞♥ ♥❤Þ t❤ø❝
✭❛✮
Zx−1
3(1 −x1/6)−1dx
✳
✭❜✮
Zx−2
3(1 + x1
3)−3dx
✳
✭❝✮
Zx−1
2(1 + x1
4)−10dx
✳
✭❞✮
Zx
p1 + 3
√x2
dx
✳
✭❡✮
Zx3(1 + 2x2)−2
3dx
✳
✭❢✮
Zdx
x4√1 + x2
✳
✭❣✮
Zdx
x2(1 + x3)5/3
✳
✭❤✮
Zdx
√x33
p1 + 4
√x3
✳
✭✐✮
Zdx
3
√x2(3
√x+ 1)3
✳
✭❥✮
Z3
√x
p3
√x+ 1dx
✳
✭❦✮
Zdx
x6√x2−1
✳
✭❧✮
Zdx
x3
√1 + x5
✳
✭♠✮
Zx7√1 + x2dx
✳
✭♥✮
Zdx
3
√1 + x3
✳
✭♦✮
Zdx
4
√1 + x4
✳
✭♣✮
Z3
√x−x3dx
✳
✶✶✳ ❚Ý♥❤ ❝➳❝ tÝ❝❤ ♣❤➞♥ s❛✉
•Zcos4xdx
✳
•Zsin5xdx
✳
•Zsin3xcos2xdx
✳
•Zcos3xsin5xdx
✳
•Zdx
sin 2x
✳
•Zdx
cos x
3
✳
•Zdx
cos x
3
✳
•Zsin x+ cos x
sin 2xdx
✳
•Zsin2x
cos6xdx
✳
•Zsin 3xcos xdx
✳
•Zsin x
3cos 2x
3dx
✳
Phép tính tích phân trên R
5
Bài tập Giải tích II
K53 -Toán tin
ĐHKHTN