Live gt1 tích phân hàm hữu tỉ thầy lam trường
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LIVE: TOÁN CAO CẤP – GT1
CHƯƠNG VI: NGUYÊN HÀM & TÍCH PHÂN BẤT ĐỊNH
TÍCH PHÂN CÁC HÀM HỮU TỈ + LƯỢNG GIÁC
A. KIẾN THỨC
III. TÍCH PHÂN CÁC HÀM SỐ HỮU TỈ
1. Tích phân các hàm số hữu tỉ đơn giản
I:
A
x − a dx = A ln x − a + c
II:
A
( x − a)
k
dx =
A
1
+c
1 − k ( x − a ) k −1
2
p
p2
Mx + N
III: 2
dx → x 2 + px + q = x + + q −
2
4
x + px + q
IV:
(x
Mx + N
2
+ px + q
)
k
dx : Tương tự III
p4
= a2
q −
4
III-IV -> Đặt
x + p = t dx = dt
2
2. Tích phân các hàm số hữu tỉ bất kỳ
+) Dạng tổng quát của hàm hữu tỉ: y = f (x) =
Pn (x)
Qm (x)
Với Pn (x), Qn (x) là các đa thức bậc n, m không nghiệm chung
+) n m → hàm hữu tỉ khơng thực sự → có thể phân tích thành tổng của một đa thức và 1 hàm
bằng cách chia tử cho mẫu
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Thầy Phạm Ngọc Lam Trường
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+) Ta chỉ xét n < m: các hàm hữu tỉ thật sự
→ Ta tiến hành phân tích y =
=
A1
( x − a)
α
+
A2
( x − a)
α −1
+ ... +
Pn (x)
Qm (x)
Aα
B1x + C1
+ ... +
( x − a)
x 2 + px + q
(
B2 x + C 2
+
) (x
β
2
+ px + q
)
β −1
+ ... +
Bβ x + C β
x 2 + px + q
Trong đó A1, A2...,B1, B2...,C1, C2,... gọi là hệ số bất định, ta có thể tìm ra bằng cách quy đồng hai vế
rồi đồng nhất hệ số hai vế.
IV. TÍCH PHÂN CÁC HÀM LƯỢNG GIÁC
1. Dạng 1. I = R ( sinx,cosx ) dx
a) Trường hợp chung: Đặt tan
+) x = 2arctant dx =
+) sinx =
x
=t
2
2dt
1+ t2
2t
1− t2
,
cosx
=
1+ t2
1+ t2
b) Các trường hợp đặc biệt:
•
R ( sinx,cosx ) = −R ( −sinx,cosx ) Đặt t = cosx .
•
R ( sinx,cosx ) = − R ( sinx, −cosx ) Đặt t = sinx
•
R ( sinx,cosx ) = R ( −sinx, −cosx ) Đặt t = tan x
2. Dạng 2: I = sinaxcosbxdx , sinaxsinbxdx, cosaxcosbxdx .
1
sin ( a + b ) x + sin ( a − b ) x
2
1
cosaxcosbx = cos ( a + b ) x + cos ( a − b ) x .
2
1
sinaxsinbx = cos ( a − b ) x − cos ( a + b ) x
2
sinaxcosbx =
→ Dùng các công thức:
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Thầy Phạm Ngọc Lam Trường
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V. DẠNG MỞ RỘNG
( )
1. Dạng: R e x dx .
+) Đặt e x = t x = lnt,dt = e x dx = t dx dx =
→ I = R (t )
dt
t
dt
là tích phân hàm hữu tỉ của t
t
2. Dạng: R ( shx,chx ) dx
shx =
ex − e−x
ex + e−x
;chx =
Tương tự dạng R e x .
2
2
( )
B. BÀI TẬP
Bài 1: Tính các tích phân sau:
a)
x
dx
x + 3x + 2
2
b)
(x
dx
2
+ 2x + 5
)
2
c)
x+2
x − 5x + 6
2
dx
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Bài 2: Tính các tích phân sau:
a)
sinx − cosx + 2
dx
1 + sinx + cosx
b)
dx
sinxcos 4 x
c)
dx
cosx 3 sin2 x
d) sinxsin ( x + y ) dx
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Bài 3: Tính các tích phân sau:
a)
e 2x
dx
1 + ex
b)
shx
ch2x
dx
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Bài 4: Tính các tích phân sau:
a)
b)
( 2x + 1) dx
( x + 2)( x + 3)
x2
( x + 1) ( x
2
)
+1
dx
c)
x2 + 2
dx
x3 + x
d)
dx
(x + 2)2 (x + 3)2
e)
x5 + x4 − 8
dx
x 3 − 4x
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Thầy Phạm Ngọc Lam Trường
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__HẾT__
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