Tính các nguyên hàm Tính các nguyên hàm. a.[r]
Trang 1ĐÁP ÁN TOÁN 12
a
(1đ) I=∫(3x 4) dx− 5 = (3x 4)6 C
18
I=∫(2x 5) d+ x = (2x 5)7 C
14
b
(1đ) I=∫sin 5x sin 2xdx=12∫(cos3x cos7x)dx−
sin 3x sin 7x C
0.5 0.5
1
I cos5x cos3xdx (cos8x cos 2x)dx
2
1 1sin8x 1sin 2x C
c
x 1 x 2
=ln x 1 2ln x 2 C− + + +
0.5 0.5
2
x 3 x 2
3ln x 3 2ln x 2 C= − + + +
d
(1.5đ)
Đặt t= x2+1 ⇒ x2= − ⇒ xdx = tdt t2 1
3
3
( )3
2
2
3
+
0.5 0.5
0.5
2x
2x 3
=
+
Đặt t= 2x 3+ ⇒ 2x t= −2 3 ⇒ dx = tdt
3
( )3
2x 3
3 2x 3 C 3
+
e
(1.5đ)
3x
I=∫(3x 2)e dx−
u 3x 2 du 3dx
1
3
⎧
⎪
⎨
1
I (3x 2)e e dx
3
1(3x 2)e3x 1e3x C
0.5
0.5 0.5
2x
I=∫(2x 1)e d+ x
u 2x 1 du 2dx
1
2
⎧
⎪
⎨
1
I (2x 1)e e d 2
1(2x 1)e2x 1e2x C
2 A(2;5;1) , B(3;1; 2)− , C( 1;8;4)− , D(1; 2;6)− A(2; 1;3)− , B( 2;3;3)− , C(1;2; 1)− , D(3;1;2)
a
(1đ) AB (1; 4; 3)JJJG= − − ; AC ( 3;3;3)JJJG= −
[AB,AC] ( 3;6; 9)JJJG JJJG = − −
AD ( 1; 7;5)= − −
JJJG
[AB,AC].ADJJJG JJJG JJJG= − ≠84 0
A, B, C, D không đồng phẳng
0.5
0.25 0.25
AB ( 4;4;0)JJJG= − ; AC ( 1;3; 4)JJJG= − − [AB,AC] ( 16; 16; 8)JJJG JJJG = − − −
AD (1;2; 1)JJJG= − [AB,AC].ADJJJG JJJG JJJG= − ≠40 0
A, B, C, D không đồng phẳng
b
(1đ) ABCD 1
6
= JJJG JJJG JJJGAD = 14 Gọi DH là chiều cao tứ diện ABCD
1
3 ∆
ABC
3V DH
S∆
=
ABC
1
2
∆ = JJJG JJJGC] = 3 14
2
DH 2 14=
0.25
0.25 0.25 0.25
= JJJG JJJG JJJG = 0 Gọi DH là chiều cao tứ diện ABCD
1
3 ∆
ABC
3V DH
S∆
=
ABC
1
2
∆ = JJJG JJJGAC] = 12 5
DH 3
=
c
(1đ) [AC,AD] (36;12;24) 12(3;1;2)JJJG JJJG = =
A(2;5;1) (ACD)
vtpt n (3;1;2)
⎧⎪
⎨
=
⎪⎩
ñi qua
G (ACD): 3x + y + 2z – 13 = 0
0.25 0.5 0.25
[AC,AD] (5; 5; 5) 5(1; 1; 1)JJJG JJJG = − − = − −
A(2; 1;3) (ACD)
vtpt n (1; 1; 1)
−
⎧⎪
⎨
= − −
⎪⎩
ñi qua
G (ACD): x – y – z = 0
Trang 2d
(1đ) Mặt cầu (S)⎧ B(3;1; 2) R−
⎨
⎩
tâm bán kính
(S) tiếp xúc mp(ACD) ⇔ R d B;(ACD)= ( )
7
R
14
=
(S) : (x 3) (y 1) (z 2)
2
0.5 0.5
R
B( 2;3;3) (S)⎧ −
⎨
⎩
tâm bán kính
(S) tiếp xúc mp(ACD) ⇔ R d B;(ACD)= ( ) 8
R 3
=
(S) : (x 2) (y 3) (z 3)
3