NGUYÊN HÀM
Dạng 39. Nguyên hàm hàm đa thức, phân thức
Câu 1. Tìm nguyên hàm của hàm số f ( x ) = 3x2 +
A.
C.
x
.
2
f ( x ) dx =x 3 +
x2
+C.
4
B.
f ( x ) dx =x 3 +
x2
+C.
2
=
x
+
+C.
2
4
5
x3
Câu 2. Tìm nguyên hàm của hàm số f ( x ) = x − 1 .
18
2
6
A.
x3
f ( x ) dx = − 1 + C .
18
C.
D.
Lời giải tham khảo
Đặt t =
x3
− 1.
18
Câu 3. Cho f ( x ) = 3 ( x + 2 ) , f ( 0 ) = 8. Hàm số y = f ( x ) là hàm số nào trong các hàm sau
2
đây?
A. f ( x ) = 2 ( x + 2 ) − 8.
3
B. f ( x ) = ( x + 2 ) + 4 .
2
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D. f ( x ) = ( x + 2 ) .
C. f ( x ) = 6 ( x + 2 ) − 4 .
1
2
2
1
.
x3
+C.
B.
f ( x ) dx = x
+C .
D.
f ( x ) dx = ln
1
4
+C.
3
x +C.
x−1
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1
A.
f ( x)dx = x + x − 1 + C .
C.
f ( x)dx =
x2
+ ln x − 1 + C .
2
1
B.
f (x)dx = 1 + ( x − 1)
D.
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Câu 7. Tìm nguyên hàm của hàm số f ( x) =
x−2
+C.
x−1
A.
f ( x)dx = ln
C.
f ( x)dx = ln x − 2 + C .
x−1
1
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Câu 8. Tìm nguyên hàm F ( x ) của hàm số f ( x ) =
x4 + 2 x3 + 1
thoả mãn F ( 1) = 2 .
x2
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A.
x3
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Câu 9. Hàm số nào sau đây không là một nguyên hàm của hàm số y =
A. y =
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Câu 10. Tìm nguyên hàm F ( x ) của hàm số f ( x ) = thoả mãn f ' ( x ) = ax+
f ( 1) = 4, f ( −1) = 2 .
b
, f ' ( 1) = 0,
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Câu 11. Tìm giá trị của tham số a để hàm số F( x) =
số f ( x) =
−6
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Câu 12. Tìm nguyên hàm của hàm số f ( x ) =
x −1
.
x2
A.
f ( x ) dx = − ln x + x + C .
B.
f ( x ) dx = ln x − x + C .
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Câu 13. Tìm nguyên hàm của hàm số f ( x) = ( 3x − 1) .
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A.
f (x)dx = 3 ( 3x − 1)
C.
f (x)dx = 18 ( 3x − 1)
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3
−2 x.
x
B.
x3
4 3
+ 3 ln x −
x .
3
3
f ( x ) dx =
x3
4 3
f ( x ) dx = − 3 ln x −
x +C .
3
3
D.
Lời giải tham khảo
C.
(
3
f ( x )dx = 1 + x 2
2
f ( x )dx =
(
2
1 + x2
3
)
)
3
2
3
2
)
3
2
+C .
+C .
Lời giải tham khảo
(
)
1
f ( x ) = 1 + x 2 xdx = 1 + x 2 2 xdx =
(
1
) (
)
(
B.
f ( x ) dx = 3 ( 3x + 2 )
3x+2 + C .
D.
f (x)dx = 2 ( 3x + 2 )
3x+2 + C .
2
3
Lời giải tham khảo
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Ta có
1
f ( x)dx = 3x+2dx = ( 3x+2) 2 dx =
)
1 + x2
2
)
+C.
2
B.
+C .
D.
f ( x ) dx =
1
3
f ( x ) dx =
1
3
1
+
x
d
1
+
x
2
2
(
1 + x2
) +C.
2
Câu 18. Tìm nguyên hàm của hàm số f ( x ) = 3 3x + 1 .
A.
f ( x) dx =(3x + 1)
C.
f ( x ) dx = 4 (3x + 1)
1
3
3x + 1 + C .
1
3
d ( 3 x + 1)
3
=
1
3
1
( 3 x + 1) d ( 3 x + 1)
3
1 ( 3 x + 1)
=
3
4
3
4
3
+C
1
3x + 1) 3 3x + 1 + C .
(
33
33
33
x5 + 14 ln 1 − x + C .
x5 + 14 ln 1 − x + C .
Lời giải tham khảo
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23
3 2
14
1
33 5
x
+
dx
=
x − 14 ln 1 − x + C .
x + 14.
dx =
B.
f ( x ) dx =2 ln
C.
f ( x) dx =2
x − 2 ln x + 1 + C.
D.
f ( x) dx =2
x + 1 + C.
x − 2 ln x + 1 + C.
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2x − 1 − ln
C.
f ( x) dx =
2x − 1 − 4 ln
(
2x − 1 + 4 + C.
D.
f ( x) dx =2
2x − 1 − ln
(
2x − 1 + 4 + C.
(
. nào sau đây là đúng?
2x − 1 + 4 + C.
)
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Câu 22. Tìm nguyên hàm của hàm số f ( x ) =
A.
f ( x ) dx =2 ln (
C.
f ( x ) dx =2 ln
x+
x+ x
.
)
B.
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Câu 23. Tìm nguyên hàm của hàm số f ( x ) =
A.
f ( x ) dx = −
C.
f ( x ) dx = −
x+x
+C.
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C.
f ( x ) dx = 2 ln x +
D.
f ( x ) dx =
1
k
1
x2 + k
x2 + k + C .
+C.
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1
trên khoảng ; + . Tính S = a + b + c .
2
B. S = 0 .
A. S = 3 .
C. S = 4 .
D. S = 2 .
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D. a = 4, b = −2, c = 1 .
C. a = −2, b = 1, c = 4 .
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B. Chỉ ( III ) .
A.Chỉ ( I ) .
D. Chỉ ( III ) và (IV).
C. Chỉ ( II ) .
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(
)
2
+C.
D.
3
1
1
f ( x ) dx = 3 x +
+C.
3
x
f ( x ) dx = 5 x
3
3
x2 + ln x +
12 5 6
x +C.
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Câu 29. Tìm nguyên hàm của hàm số f ( x ) =
x
a2 + x2
.
A.
f ( x ) dx =
1 + x2 + C .
B.
f ( x ) dx = ln a + x
C.
f ( x ) dx =
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Dạng 41. Nguyên hàm hàm lượng giác
Câu 30. Tìm nguyên hàm của hàm số f ( x ) = sin 2 x .
A.
f ( x ) dx = 2 −
sin 2x
+C .
2
B.
f ( x ) dx = 2 +
sin
2
xdx =
1 − cos 2x
x sin 2x
dx = −
+C.
2
2
4
Câu 31. Tìm nguyên hàm của hàm số f ( x ) = sin ( 2 x + 1) .
−1
cos(2x + 1) + C .
2
A.
f ( x ) dx = cos(2x + 1) + C .
B.
f ( x ) dx =
C.
1
C. F( x) = x − cos 3x − .
3
6
1
D. F( x) = x − cos 3x + .
3
6
Lời giải tham khảo
1
F ( x ) = x − cos 3x − .
3
6
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Câu 33. Tìm nguyên hàm F ( x ) của hàm số f ( x ) = 2 x 3 − 3x 2 + 1 − sin 2 x thoả mãn
F (0) = 1 .
A. F ( x ) = 2
2
x4
x3
1
1
D. F ( x ) = 2 − 3 + x + .cos 2 x − .
4
3
2
2
Lời giải tham khảo
(
F ( x)
)
= 2 x3 − 3x2 + 1 − sin 2 x dx = 2
Vì F ( 0 ) = 1 nên
x4
x3
1
− 3 + x + .cos 2 x + C
4
3
C.
e
D.
x
dx = e x + C .
1
sin
2
x
dx = − tan x + C .
Lời giải tham khảo
1
sin
2
x
2
2
x
dx = − cot x + C
Câu 37. Tìm nguyên hàm của hàm số f ( x ) = x cos x .
A.
f ( x ) dx =x sin x – cos x + C .
B.
f ( x ) dx = − x sin x – cos x + C .
C.
f ( x ) dx =x sin x + cos x + C .
D.
f ( x ) dx = − x sin x + cos x + C .
Lời giải tham khảo
Đặt u = x, dv = cos xdx; ta chọn du = dx, v = sin x.
Do đó I = x sin x + sin xdx = x sin x − cos x + C .
Câu 38. Tìm nguyên hàm của hàm số f ( x ) = sin 3x.cos5x .
A.
Lời giải tham khảo
f (x)dx = 2 ( sin 8x − sin 2x )dx = 4 cos2x − 16 cos8x + C .
1
1
1
Câu 39. Tìm nguyên hàm của hàm số f ( x ) =
A.
1
1
1
sin cos .
2
x
x
x
f ( x ) dx = 4 cos x + C.
1
f ( x ) dx = 4 cos x + C.
1
1
Đặt t =
Câu 40. Tìm nguyên hàm F ( x ) của hàm số f ( x ) = −
A. – tan x .
B. 1 – tan x .
1
thoả mãn F ( 0 ) = 1.
cos2 x
C. 1 + tan x .
D. tan x − 1 .
Lời giải tham khảo
Chọn đáp án B.
F ( x) = −
1
dx = − tan x + C. F ( 0 ) = 1 nên C = 1.
cos2 x
BÀI TẬP TỰ LUYỆN ✓
Câu 41. Tìm nguyên hàm của hàm số f ( x ) =
x
.
cos2 x
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Câu 42. Tìm nguyên hàm của hàm số f ( x ) =
sin 3 x
.
cos 4 x
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A.
f ( x ) dx = 3 cos
1
3
x
1
3
1
3
x
x
−
−
1
+C .
cos x
1
+C.
cos2 x
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A.
f ( x ) dx = cot x + tan x + C.
B.
f ( x ) dx = − cot x − tan x + C.
C.
f ( x ) dx = − cot x + tan x + C.
D.
f ( x ) dx = cot x − tan x + C.
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f ( x ) dx = − 2 tan x + 4 + C.
C.
f ( x ) dx = − 2 tan x − 4 + C.
1
1
1
( cos x + sin x )
2
.
B.
f ( x ) dx = 2 tan x − 4 + C.
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Câu 45. Cho I =
A. A = B =
s inx
cos x − s inx
dx = A + B
dx . Tính giá trị A , B .
cos x + s inx
cos
x
+
s
in
x
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Câu 47. Xét các mệnh sau đây:
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Câu
2
1
2
2
2
tan x +
tan x −
.
2
2
2
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B. F ( x ) = e x
2
C. F ( x ) = e x
2
D. F ( x ) = e x
tan x −
.
2
2
2
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( )
Câu 50. Tìm nguyên hàm của hàm số f ( x ) = xcos x 2 .
A.
f ( x ) dx = 2 s inx + C .
B.
f ( x ) dx = − 2 s inx + C .
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Câu 51. Tìm nguyên hàm của hàm số f ( x ) = x sin 2 x .
A.
1
f ( x ) dx = 2 x
2
− x sin 2 x −
1
cos 2 x + C .
2
− x sin 2 x +
1
cos 2 x + C .
2
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D.
1
dx =
1
( + 1)
( ax + b )
+1
+C.
e
ax+b
1
dx = e ax+b + C .
a
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A.
f ( x ) dx = cos 2x − sin x + C .
B.
f ( x ) dx = − 2 cos 2x + sin x + C .
C.
f ( x ) dx = − cos 2x + sin x + C .
D.
f ( x ) dx = sin
1
2
x + sin x + C .
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2
f ( x ) dx =
x3
1
f ( x ) dx = + ln | x | − cos 2 x + C .
3
2
1
− sin 2x .
x
B.
D.
x3
1
− ln | x | + cos 2 x + C .
3
2
f ( x ) dx =
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