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Worksheets

nguyên hàm toán 12

Total questions: 91

Worksheet time: 1hrs 13mins

Name
Class
Date
1.

∫sin⁡xdx =\int^{ }\sin xdx\ =  

a)

cos⁡x+C\cos x+C  

b)

−cos⁡x+C-\cos x+C  

c)

sin⁡x+C\sin x+C  

d)

−sin⁡x+C-\sin x+C  

2.

∫cos⁡xdx=\int_{ }^{ }\cos xdx=  

a)

−cos⁡x+C-\cos x+C  

b)

cos⁡x+C\cos x+C  

c)

−sin⁡x+C-\sin x+C  

d)

sin⁡x+C\sin x+C  

3.

∫1cos⁡2xdx=\int\frac{1}{\cos^2x}dx=  

a)

tan⁡x+C\tan x+C  

b)

−tan⁡x+C-\tan x+C  

c)

cot⁡x+C\cot x+C  

d)

−cot⁡x+C-\cot x+C  

4.

∫1sin⁡2xdx=\int_{ }^{ }\frac{1}{\sin^2x}dx=  

a)

tan⁡x+C\tan x+C  

b)

cot⁡x+C\cot x+C  

c)

−tan⁡x+C-\tan x+C  

d)

−cot⁡x+C-\cot x+C  

5.

∫1xdx =\int_{ }\frac{1}{x}dx\ =  

a)

ln⁡∣x∣+C\ln\left|x\right|+C  

b)

ln⁡x+C\ln x+C  

6.

∫1ax+bdx=\int_{ }^{ }\frac{1}{ax+b}dx=  

a)

1aln⁡∣ax+b∣+C\frac{1}{a}\ln\left|ax+b\right|+C  

b)

ln⁡∣ax+b∣+C\ln\left|ax+b\right|+C  

7.

∫exdx=\int e^xdx=  

a)

ex+Ce^x+C  

b)

−ex+C-e^x+C  

8.

∫e(ax+b)dx =\int e^{\left(ax+b\right)}dx\ =  

a)

1ae(ax+b)+C\frac{1}{a}e^{\left(ax+b\right)}+C  

b)

−1ae(ax+b)+C-\frac{1}{a}e^{\left(ax+b\right)}+C  

9.

∫xαdx=    ,(α≠−1)\int_{ }^{ }x^{\alpha}dx=\ \ \ \ ,\left(\alpha\ne-1\right)  

a)

αx(α−1)+C\alpha x^{\left(\alpha-1\right)}+C  

b)

x(α+1)α+1\frac{x^{\left(\alpha+1\right)}}{\alpha+1}  +C

10.

∫sin⁡(ax+b)dx =\int_{ }\sin\left(ax+b\right)dx\ =  

a)

1acos⁡(ax+b)+C\frac{1}{a}\cos\left(ax+b\right)+C  

b)

−1acos⁡(ax+b)+C-\frac{1}{a}\cos\left(ax+b\right)+C  

11.

Cho hàm số ff  xác định trên  KK  . Hàm số  FF  dược gọi là nguyên hàm của  ff  trên  KK  nếu

a)

f′(x)=F(x),∀x∈Kf'\left(x\right)=F\left(x\right),\forall x\in K  

b)

F′(x)=f(x),∀x∈KF'\left(x\right)=f\left(x\right),\forall x\in K  

12.

Chọn các khẳng định đúng
Nếu  f,gf,g  là hai hàm số liên tục trên  KK  thì

a)

∫[f(x)+g(x)]dx=∫f(x)dx+∫g(x)dx\int_{ }\left[f\left(x\right)+g\left(x\right)\right]dx=\int f\left(x\right)dx+\int g\left(x\right)dx  

b)

(k≠0)\left(k\ne0\right)  

∫kf(x)dx=k∫f(x)dx\int kf\left(x\right)dx=k\int f\left(x\right)dx  

c)

∫[f(x).g(x)]dx=∫f(x)dx.∫g(x)dx\int_{ }\left[f\left(x\right).g\left(x\right)\right]dx=\int f\left(x\right)dx.\int g\left(x\right)dx  

13.

∫cos⁡(ax+b)dx =\int_{ }\cos\left(ax+b\right)dx\ =  

a)

1asin⁡(ax+b)+C\frac{1}{a}\sin\left(ax+b\right)+C  

b)

−1asin⁡(ax+b)+C-\frac{1}{a}\sin\left(ax+b\right)+C  

14.

∫axdx=      (0<a≠1)\int_{ }^{ }a^xdx=\ \ \ \ \ \ \left(0<a\ne1\right)  

a)

axln⁡a+C\frac{a^x}{\ln a}+C  

b)

axln⁡aa^x\ln a  

15.

∫dx=∫1dx=\int_{ }^{ }dx=\int_{ }^{ }1dx=  

a)

x2+Cx^2+C

b)

x+Cx+C

16.

Tích phân

∫1eln⁡xxdx\int_1^e\frac{\ln x}{x}dx  bằng

a)

−3 -\sqrt{3\ }  

b)

11  

c)

ln⁡2\ln2  

d)

12\frac{1}{2}  

17.

Tıˊnh∫e2xdxTính\int e^{2x}dx  

a)

2.e2x+C2.e^{2x}+C  

b)

e2x+20212+C\frac{e^{2x}+2021}{2}+C  

c)

ex2+C\frac{e^x}{2}+C  

d)

−e2x2+C-\frac{e^{2x}}{2}+C  

18.

∫tan⁡x dx\int\tan x\ dx  

Tìm

a)

1cos⁡2x+C\frac{1}{\cos^2x}+C  

b)

−1cos⁡2x+C-\frac{1}{\cos^2x}+C  

c)

−ln⁡∣cos⁡x∣ +C-\ln\left|\cos x\right|\ +C  

d)

ln⁡∣cos⁡x∣ +C\ln\left|\cos x\right|\ +C  

19.

  Ha~y\ \ Hãy  Tính ∫13xdx\int_{ }^{ }\frac{1}{3x}^{ }dx  

a)

3.ln(x) + C

b)

ln⁡(x)3+C\frac{\ln\left(x\right)}{3}+C  

c)

ln⁡(x)3\frac{\ln\left(x\right)}{3}  

d)

ln⁡∣2021x∣3+C\frac{\ln\left|2021x\right|}{3}+C  

20.

Tıˊnh∫sin⁡x dx Tính\int_{ }^{ }\sin x\ dx\  

a)

Cosx +C

b)

- Cosx + C

c)

- Sinx + C

d)

Sinx + C

21.

f(x)=x3−3x2+5f\left(x\right)=x^3-3x^2+5  

Họ nguyên hàm của hàm số là

a)

3x2−6x3x^2-6x  

b)

3x2−6x+C3x^2-6x+C  

c)

x44−x3+5x+C\frac{x^4}{4}-x^3+5x+C  

d)

x4−x3+5x+Cx^4-x^3+5x+C  

22.

∫sin⁡xdx =\int^{ }\sin xdx\ =  

a)

cos⁡x+C\cos x+C  

b)

−cos⁡x+C-\cos x+C  

c)

sin⁡x+C\sin x+C  

d)

−sin⁡x+C-\sin x+C  

23.

∫cos⁡xdx=\int_{ }^{ }\cos xdx=  

a)

−cos⁡x+C-\cos x+C  

b)

cos⁡x+C\cos x+C  

c)

−sin⁡x+C-\sin x+C  

d)

sin⁡x+C\sin x+C  

24.

∫1cos⁡2xdx=\int\frac{1}{\cos^2x}dx=  

a)

tan⁡x+C\tan x+C  

b)

−tan⁡x+C-\tan x+C  

c)

cot⁡x+C\cot x+C  

d)

−cot⁡x+C-\cot x+C  

25.

∫1sin⁡2xdx=\int_{ }^{ }\frac{1}{\sin^2x}dx=  

a)

tan⁡x+C\tan x+C  

b)

cot⁡x+C\cot x+C  

c)

−tan⁡x+C-\tan x+C  

d)

−cot⁡x+C-\cot x+C  

26.

∫1ax+bdx=\int_{ }^{ }\frac{1}{ax+b}dx=  

a)

1aln⁡∣ax+b∣+C\frac{1}{a}\ln\left|ax+b\right|+C  

b)

ln⁡∣ax+b∣+C\ln\left|ax+b\right|+C  

27.

∫e(ax+b)dx =\int e^{\left(ax+b\right)}dx\ =  

a)

1ae(ax+b)+C\frac{1}{a}e^{\left(ax+b\right)}+C  

b)

−1ae(ax+b)+C-\frac{1}{a}e^{\left(ax+b\right)}+C  

28.

∫sin⁡(ax+b)dx =\int_{ }\sin\left(ax+b\right)dx\ =  

a)

1acos⁡(ax+b)+C\frac{1}{a}\cos\left(ax+b\right)+C  

b)

−1acos⁡(ax+b)+C-\frac{1}{a}\cos\left(ax+b\right)+C  

29.

Cho hàm số ff  xác định trên  KK  . Hàm số  FF  dược gọi là nguyên hàm của  ff  trên  KK  nếu

a)

f′(x)=F(x),∀x∈Kf'\left(x\right)=F\left(x\right),\forall x\in K  

b)

F′(x)=f(x),∀x∈KF'\left(x\right)=f\left(x\right),\forall x\in K  

30.

∫axdx=      (0<a≠1)\int_{ }^{ }a^xdx=\ \ \ \ \ \ \left(0<a\ne1\right)  

a)

axln⁡a+C\frac{a^x}{\ln a}+C  

b)

axln⁡aa^x\ln a  

31.

∫exdx=\int e^xdx=  

a)

ex+Ce^x+C  

b)

−ex+C-e^x+C  

c)

ex+x+Ce^x+x+C  

d)

ex+1x+1+C\frac{e^x+1}{x+1}+C  

32.

 Nguyên hàm của hàm số f(x)=x2−3xf\left(x\right)=x^2-3x là ? 

a)

F(x)=x33−3x22+CF\left(x\right)=\frac{x^3}{3}-\frac{3x^2}{2}+C  

b)

F(x)=x33+3x22+CF\left(x\right)=\frac{x^3}{3}+\frac{3x^2}{2}+C  

c)

F(x)=x3−3x2+CF\left(x\right)=x^3-3x^2+C  

d)

F(x)=x33−3x22F\left(x\right)=\frac{x^3}{3}-\frac{3x^2}{2}  

33.

Nguyên hàm của hàm số f(x)=4x3+1f\left(x\right)=4x^3+1  là ?

a)

F(x)=x4−x+CF\left(x\right)=x^4-x+C  

b)

F(x)=x4+x+CF\left(x\right)=x^4+x+C  

c)

F(x)=4x4+x+CF\left(x\right)=4x^4+x+C  

d)

F(x)=12x2+CF\left(x\right)=12x^2+C  

34.

nguyên hàm của hàm số  f(x)=sin⁡x−2cos⁡xf\left(x\right)=\sin x-2\cos x là ? 

a)

F(x)=sin⁡x+cos⁡x+CF\left(x\right)=\sin x+\cos x+C  

b)

F(x)=cos⁡x+2sin⁡x+CF\left(x\right)=\cos x+2\sin x+C  

c)

F(x)=−cos⁡x−2sin⁡x+CF\left(x\right)=-\cos x-2\sin x+C  

d)

F(x)=cos⁡x+2sin⁡x+CF\left(x\right)=\cos x+2\sin x+C  

35.

Nguyên hàm ∫3(x−2)2dx\int_{ }^{ }3\left(x-2\right)^2dx  là ?

a)

x3−6x2+12xx^3-6x^2+12x  

b)

x3−6x2+12x+Cx^3-6x^2+12x+C  

c)

x3+6x2+12x+Cx^3+6x^2+12x+C  

d)

6x−12+C6x-12+C  

36.

Cho F(x) là một nguyên hàm của hàm số  f(x)=4x3−2x+1f\left(x\right)=4x^3-2x+1  , biết F(2) = 14. Tính F(x) ? 

a)

F(x)=x4−x2+xF\left(x\right)=x^4-x^2+x  

b)

F(x)=x4−x2+x+CF\left(x\right)=x^4-x^2+x+C  

c)

F(x)=x4−x2+x+2F\left(x\right)=x^4-x^2+x+2  

d)

F(x)=12x2−2+CF\left(x\right)=12x^2-2+C  

37.

∫xndx=?\int_{ }^{ }x^ndx=?  

a)

xnn\frac{x^n}{n}  

b)

x(n+1)n+1\frac{x^{\left(n+1\right)}}{n+1}  

c)

x(n+1)n+1+C\frac{x^{\left(n+1\right)}}{n+1}+C  

d)

xnn+C\frac{x^n}{n}+C  

38.

Họ các nguyên hàm của f(x)=x2+4x3f\left(x\right)=x^2+4x^3  là:

a)

2x+12x2+C2x+12x^2+C  

b)

x33+x4+C\frac{x^3}{3}+x^4+C  

c)

x3+x44+Cx^3+\frac{x^4}{4}+C  

d)

x3+x24+C\frac{x}{3}+\frac{x^2}{4}+C  

39.

∫axdx=?\int_{ }^{ }a^xdx=?  

a)

ax+Ca^x+C  

b)

ax.ln⁡a+Ca^x.\ln a+C  

c)

ex+Ce^x+C  

d)

axln⁡a+C\frac{a^x}{\ln a}+C  

40.

∫(5x+1)dx=?\int_{ }^{ }\left(5^x+1\right)dx=?  

a)

5x+x+C5^x+x+C  

b)

5xln⁡5+x+C\frac{5^x}{\ln5}+x+C  

c)

5xln⁡5+2x+C\frac{5^x}{\ln5}+2x+C  

d)

5x+2x+C5^x+2x+C  

41.

∫sin⁡xdx=?\int_{ }^{ }\sin xdx=?  

a)

−cos⁡x+C-\cos x+C  

b)

cos⁡x+C\cos x+C  

c)

cos⁡2x+C\cos^2x+C  

d)

−cos⁡2x+C-\cos^2x+C  

42.

∫(3cos⁡x−sin⁡(3x−2))dx=?\int_{ }^{ }\left(3\cos x-\sin\left(3x-2\right)\right)dx=?  

a)

3cos⁡x−sin⁡(3x−2)+C3\cos x-\sin\left(3x-2\right)+C  

b)

3sin⁡x−cos⁡(3x−2)+C3\sin x-\cos\left(3x-2\right)+C  

c)

3sin⁡x+13cos⁡(3x−2)+C3\sin x+\frac{1}{3}\cos\left(3x-2\right)+C  

d)

3cos⁡x+13sin⁡(3x−2)+C3\cos x+\frac{1}{3}\sin\left(3x-2\right)+C  

43.

Họ các nguyên hàm của f(x)=1ax+bf\left(x\right)=\frac{1}{ax+b}  là:

a)

a.ln⁡∣ax+b∣a.\ln\left|ax+b\right|  

b)

a.ln⁡∣ax+b∣+Ca.\ln\left|ax+b\right|+C  

c)

1a.ln⁡∣ax+b∣\frac{1}{a}.\ln\left|ax+b\right|  

d)

1a.ln⁡∣ax+b∣+C\frac{1}{a}.\ln\left|ax+b\right|+C  

44.

∫(13x+4+5cos⁡x)dx\int_{ }^{ }\left(\frac{1}{3x+4}+5\cos x\right)dx  

a)

3ln⁡∣3x+4∣+sin⁡x+C3\ln\left|3x+4\right|+\sin x+C  

b)

13ln⁡∣3x+4∣+5sin⁡x+C\frac{1}{3}\ln\left|3x+4\right|+5\sin x+C  

c)

13ln⁡∣3x+4∣+5cos⁡x+C\frac{1}{3}\ln\left|3x+4\right|+5\cos x+C  

d)

3ln⁡∣3x+4∣+cos⁡x+C3\ln\left|3x+4\right|+\cos x+C  

45.

∫exdx=?\int_{ }^{ }e^xdx=?  

a)

exe^x  

b)

ex2e^{x^2}  

c)

ex+Ce^x+C  

d)

ex2+Ce^{x^2}+C  

46.

   ∫f(x)dx=?\int_{ }^{ }f\left(x\right)dx=?   biết f(x)f\left(x\right)  là đạo hàm của 5sin⁡2x+3x5\sin^2x+3x  .

a)

10sin⁡x.cos⁡x+3+C10\sin x.\cos x+3+C  

b)

10sin⁡x+3+C10\sin x+3+C  

c)

5sin⁡2x+3x5\sin^2x+3x  

d)

5sin⁡2x+3x+C5\sin^2x+3x+C  

47.

∫1cos⁡2xdx=\int\frac{1}{\cos^2x}dx=  

a)

tan⁡x+C\tan x+C  

b)

−tan⁡x+C-\tan x+C  

c)

cot⁡x+C\cot x+C  

d)

−cot⁡x+C-\cot x+C  

48.

∫1sin⁡2xdx=\int_{ }^{ }\frac{1}{\sin^2x}dx=  

a)

tan⁡x+C\tan x+C  

b)

cot⁡x+C\cot x+C  

c)

−tan⁡x+C-\tan x+C  

d)

−cot⁡x+C-\cot x+C  

49.

∫cos⁡xdx=\int_{ }^{ }\cos xdx=  

a)

−cos⁡x+C-\cos x+C  

b)

cos⁡x+C\cos x+C  

c)

−sin⁡x+C-\sin x+C  

d)

sin⁡x+C\sin x+C  

50.

∫1xdx =\int_{ }\frac{1}{x}dx\ =  

a)

ln⁡∣x∣+C\ln\left|x\right|+C  

b)

ln⁡x+C\ln x+C  

51.

∫1ax+bdx=\int_{ }^{ }\frac{1}{ax+b}dx=  

a)

1aln⁡∣ax+b∣+C\frac{1}{a}\ln\left|ax+b\right|+C  

b)

ln⁡∣ax+b∣+C\ln\left|ax+b\right|+C  

52.

∫exdx=\int e^xdx=  

a)

ex+Ce^x+C  

b)

−ex+C-e^x+C  

53.

∫e(ax+b)dx =\int e^{\left(ax+b\right)}dx\ =  

a)

1ae(ax+b)+C\frac{1}{a}e^{\left(ax+b\right)}+C  

b)

−1ae(ax+b)+C-\frac{1}{a}e^{\left(ax+b\right)}+C  

54.

∫sin⁡(ax+b)dx =\int_{ }\sin\left(ax+b\right)dx\ =  

a)

1acos⁡(ax+b)+C\frac{1}{a}\cos\left(ax+b\right)+C  

b)

−1acos⁡(ax+b)+C-\frac{1}{a}\cos\left(ax+b\right)+C  

55.

∫sin⁡xdx =\int^{ }\sin xdx\ =  

a)

cos⁡x+C\cos x+C  

b)

−cos⁡x+C-\cos x+C  

c)

sin⁡x+C\sin x+C  

d)

−sin⁡x+C-\sin x+C  

56.

Cho hàm số ff  xác định trên  KK  . Hàm số  FF  dược gọi là nguyên hàm của  ff  trên  KK  nếu

a)

f′(x)=F(x),∀x∈Kf'\left(x\right)=F\left(x\right),\forall x\in K  

b)

F′(x)=f(x),∀x∈KF'\left(x\right)=f\left(x\right),\forall x\in K  

57.

∫xαdx=    ,(α≠−1)\int_{ }^{ }x^{\alpha}dx=\ \ \ \ ,\left(\alpha\ne-1\right)  

a)

αx(α−1)+C\alpha x^{\left(\alpha-1\right)}+C  

b)

x(α+1)α+1\frac{x^{\left(\alpha+1\right)}}{\alpha+1}  +C

58.

∫cos⁡(ax+b)dx =\int_{ }\cos\left(ax+b\right)dx\ =  

a)

1asin⁡(ax+b)+C\frac{1}{a}\sin\left(ax+b\right)+C  

b)

−1asin⁡(ax+b)+C-\frac{1}{a}\sin\left(ax+b\right)+C  

59.

∫axdx=      (0<a≠1)\int_{ }^{ }a^xdx=\ \ \ \ \ \ \left(0<a\ne1\right)  

a)

axln⁡a+C\frac{a^x}{\ln a}+C  

b)

axln⁡aa^x\ln a  

60.

∫dx=∫1dx=\int_{ }^{ }dx=\int_{ }^{ }1dx=  

a)

x2+Cx^2+C

b)

x+Cx+C

61.

Chọn các khẳng định đúng
Nếu  f,gf,g  là hai hàm số liên tục trên  KK  thì

a)

∫[f(x)+g(x)]dx=∫f(x)dx+∫g(x)dx\int_{ }\left[f\left(x\right)+g\left(x\right)\right]dx=\int f\left(x\right)dx+\int g\left(x\right)dx  

b)

(k≠0)\left(k\ne0\right)  

∫kf(x)dx=k∫f(x)dx\int kf\left(x\right)dx=k\int f\left(x\right)dx  

c)

∫[f(x).g(x)]dx=∫f(x)dx.∫g(x)dx\int_{ }\left[f\left(x\right).g\left(x\right)\right]dx=\int f\left(x\right)dx.\int g\left(x\right)dx  

62.

Tính

∫(x2+2x)dx\int_{ }\left(\frac{\sqrt{x}}{2}+\frac{2}{\sqrt{x}}\right)dx  

a)

I=13x3−4x+CI=\frac{1}{3}\sqrt{x^3}-4\sqrt{x}+C  

b)

I=−13x3−4x+CI=-\frac{1}{3}\sqrt{x^3}-4\sqrt{x}+C  

c)

I=13x3+4x+CI=\frac{1}{3}\sqrt{x^3}+4\sqrt{x}+C  

d)

I=13x3+CI=\frac{1}{3}\sqrt{x^3}+C  

63.

f(x)=x3−3x2+5f\left(x\right)=x^3-3x^2+5  

Họ nguyên hàm của hàm số là

a)

3x2−6x3x^2-6x  

b)

3x2−6x+C3x^2-6x+C  

c)

x44−x3+5x+C\frac{x^4}{4}-x^3+5x+C  

d)

x4−x3+5x+Cx^4-x^3+5x+C  

64.

F(x)=mx3+(3m+2n)x2+2x−3F\left(x\right)=mx^3+\left(3m+2n\right)x^2+2x-3  

là một nguyên hàm của hàm số 

f(x)=3x2+10x+2f\left(x\right)=3x^2+10x+2  . Tính giá trị m.n.

a)

2

b)

3

c)

4

d)

1

65.

f(x)=(2x+1)2019f\left(x\right)=\left(2x+1\right)^{2019}  

Họ nguyên hàm của hàm số là

a)

(2x+1)20202020+C\frac{\left(2x+1\right)^{2020}}{2020}+C  

b)

(2x+1)20204040+C\frac{\left(2x+1\right)^{2020}}{4040}+C  

c)

(2x+1)20201010+C\frac{\left(2x+1\right)^{2020}}{1010}+C  

d)

2019(2x+1)2018+c2019\left(2x+1\right)^{2018}+c  

66.

f(x)=3x−2f\left(x\right)=\sqrt{3x-2}  

Nguyên hàm của hàm số là

a)

23(3x+2)(3x+2)+C\frac{2}{3}\left(3x+2\right)\sqrt{\left(3x+2\right)}+C  

b)

13(3x+2)(3x+2)+C\frac{1}{3}\left(3x+2\right)\sqrt{\left(3x+2\right)}+C  

c)

29(3x+2)(3x+2)+C\frac{2}{9}\left(3x+2\right)\sqrt{\left(3x+2\right)}+C  

d)

32⋅13x+2+C\frac{3}{2}\cdot\frac{1}{\sqrt{3x+2}}+C  

67.

∫x(x2+2)10dx\int_{ }x\left(x^2+2\right)^{10}dx  là

a)

2(x2+2)1111+C\frac{2\left(x^2+2\right)^{11}}{11}+C  

b)

(x2+2)1122+C\frac{\left(x^2+2\right)^{11}}{22}+C  

c)

(x2+2)1111+C\frac{\left(x^2+2\right)^{11}}{11}+C  

d)

22(x2+2)11+C22\left(x^2+2\right)^{11}+C  

68.

f(x)=1x+1x3f\left(x\right)=\frac{1}{x}+\frac{1}{x^3}  có họ nguyên hàm là

a)

ln⁡x+4x4+C\ln x+\frac{4}{x^4}+C  

b)

ln⁡x+12x2+C\ln x+\frac{1}{2x^2}+C  

c)

ln⁡∣x∣−12x2+C\ln\left|x\right|-\frac{1}{2x^2}+C  

d)

ln⁡∣x∣−3x4+C\ln\left|x\right|-\frac{3}{x^4}+C  

69.

∫4ln⁡x+1xdx\int\frac{\sqrt{4\ln x+1}}{x}dx  

a)

16(4ln⁡x+1)3+C\frac{1}{6}\sqrt{\left(4\ln x+1\right)^3}+C  

b)

12(4ln⁡x+1)3+C\frac{1}{2}\sqrt{\left(4\ln x+1\right)^3}+C  

c)

32(4ln⁡x+1)3+C\frac{3}{2}\sqrt{\left(4\ln x+1\right)^3}+C  

d)

13(4ln⁡x+1)3+C\frac{1}{3}\sqrt{\left(4\ln x+1\right)^3}+C  

70.

∫cos⁡xsin⁡x+2dx\int\frac{\cos x}{\sin x+2}dx  là

a)

ln⁡(sin⁡x+2)+C\ln\left(\sin x+2\right)+C  

b)

−ln⁡∣sin⁡x+2∣+C-\ln\left|\sin x+2\right|+C  

c)

ln⁡(cos⁡x+2)+C\ln\left(\cos x+2\right)+C  

d)

−ln⁡∣cos⁡x+2∣+C-\ln\left|\cos x+2\right|+C  

71.

y=−cos⁡2x⋅sin⁡xy=-\cos^2x\cdot\sin x  
Một nguyên hàm của hàm số trên là

a)

23cos⁡2x⋅sin⁡x+1\frac{2}{3}\cos^2x\cdot\sin x+1  

b)

23cos⁡3x+7\frac{2}{3}\cos^3x+7  

c)

23sin⁡2x⋅cos⁡x+5\frac{2}{3}\sin^2x\cdot\cos x+5  

d)

13cos⁡3x+2\frac{1}{3}\cos^3x+2  

72.

f(x)=3x2+2ex−1f\left(x\right)=3x^2+2e^x-1  

Biết F(x) là nguyên hàm của hàm số trên và F(0) =1. Chọn khẳng định đúng.

a)

F(x)=x3+2ex−x−1F\left(x\right)=x^3+2e^x-x-1  

b)

F(x)=x3+2ex−xF\left(x\right)=x^3+2e^x-x  

c)

F(x)=x3+2ex−x−1F\left(x\right)=x^3+\frac{2}{e^x}-x-1  

d)

F(x)=x3−2ex−x+2F\left(x\right)=x^3-2e^x-x+2  

73.

∫ 3x2+2x+1x+1dx\int\ \frac{3x^2+2x+1}{x+1}dx  là

a)

32x2+x+ln⁡∣x+1∣+C\frac{3}{2}x^2+x+\ln\left|x+1\right|+C  

b)

32x2−x−2ln⁡∣x+1∣+C\frac{3}{2}x^2-x-2\ln\left|x+1\right|+C  

c)

32x2−2x+ln⁡∣x+1∣+C\frac{3}{2}x^2-2x+\ln\left|x+1\right|+C  

d)

32x2−x+2ln⁡∣x+1∣+C\frac{3}{2}x^2-x+2\ln\left|x+1\right|+C  

74.

f(x)=6x+93x+5f\left(x\right)=\frac{6x+9}{3x+5}  

F(x) là nguyên hàm của hàm số trên và 3.F(0) = 0. Tìm F(x)

a)

2x−ln⁡∣3x+5∣+ln⁡52x-\ln\left|3x+5\right|+\ln5

b)

2x−13ln⁡∣3x+5∣+ln⁡52x-\frac{1}{3}\ln\left|3x+5\right|+\ln5  

c)

2x−13ln⁡∣3x+5∣+13ln⁡52x-\frac{1}{3}\ln\left|3x+5\right|+\frac{1}{3}\ln5  

d)

2x−ln⁡∣3x+5∣+13ln⁡52x-\ln\left|3x+5\right|+\frac{1}{3}\ln5  

75.

f(x)=4x(1+ln⁡x)f\left(x\right)=4x\left(1+\ln x\right)  họ nguyên hàm của hàm số là

a)

2x2ln⁡x+3x22x^2\ln x+3x^2  

b)

2x2ln⁡x+x22x^2\ln x+x^2  

c)

2x2ln⁡x+3x2+C2x^2\ln x+3x^2+C  

d)

2x2ln⁡x+x2+C2x^2\ln x+x^2+C  

76.

∫(sin⁡x+xln⁡x)dx\int\left(\sin x+x\ln x\right)dx  

a)

−cos⁡x+ln⁡x+C-\cos x+\ln x+C  

b)

−cos⁡x+x22ln⁡x+x24+C-\cos x+\frac{x^2}{2}\ln x+\frac{x^2}{4}+C  

c)

−cos⁡x+x22ln⁡x−x24+C-\cos x+\frac{x^2}{2}\ln x-\frac{x^2}{4}+C  

d)

−cos⁡x+ln⁡x+x2+C-\cos x+\ln x+x^2+C  

77.

∫sin⁡xdx =\int^{ }\sin xdx\ =  

a)

cos⁡x+C\cos x+C  

b)

−cos⁡x+C-\cos x+C  

c)

sin⁡x+C\sin x+C  

d)

−sin⁡x+C-\sin x+C  

78.

∫sin⁡(ax+b)dx =\int_{ }\sin\left(ax+b\right)dx\ =  

a)

1acos⁡(ax+b)+C\frac{1}{a}\cos\left(ax+b\right)+C  

b)

−1acos⁡(ax+b)+C-\frac{1}{a}\cos\left(ax+b\right)+C  

79.

Giả sử F(x) là một nguyên hàm của hàm số f(x)=exf\left(x\right)=e^x , biết F(0)=4. Tính F(x) 

a)

F(x)=ex+2F\left(x\right)=e^x+2  

b)

F(x)=ex+3F\left(x\right)=e^x+3  

c)

F(x)=ex+4F\left(x\right)=e^x+4  

d)

F(x)=ex+1F\left(x\right)=e^x+1  

80.

Nguyên hàm của 2x(1+3x3)+C2x\left(1+3x^3\right)+C  là

a)

x2(x+x3)+Cx^2\left(x+x^3\right)+C  

b)

x2(1+3x2)+Cx^2\left(1+3x^2\right)+C  

c)

2x(x+x3)+C2x\left(x+x^3\right)+C  

d)

x2(1+6x35)+Cx^2\left(1+\frac{6x^3}{5}\right)+C  

81.

∫(5x+x3)dx\int\left(\frac{5}{x}+\sqrt{x^3}\right)dx  là

a)

5ln⁡∣x∣−25x5+C5\ln\left|x\right|-\frac{2}{5}\sqrt{x^5}+C  

b)

−5ln⁡∣x∣+25x5+C-5\ln\left|x\right|+\frac{2}{5}\sqrt{x^5}+C  

c)

−5ln⁡∣x∣−25x5+C-5\ln\left|x\right|-\frac{2}{5}\sqrt{x^5}+C  

d)

5ln⁡∣x∣+25x5+C5\ln\left|x\right|+\frac{2}{5}\sqrt{x^5}+C  

82.

−cos⁡x+C-\cos x+C là kết quả của nguyên hàm nào sau đây?

a)

∫4dx\int_{ }^{ }4dx

b)

∫−cos⁡xdx\int_{ }^{ }-\cos xdx

c)

∫cos⁡xdx\int_{ }^{ }\cos xdx

d)

∫sin⁡xdx\int_{ }^{ }\sin xdx

e)

∫−sin⁡xdx\int_{ }^{ }-\sin xdx

83.

x33+C\frac{x^3}{3}+C là kết quả của nguyên hàm nào sau đây?

a)

∫x2dx\int_{ }^{ }x^2dx

b)

∫x3dx\int_{ }^{ }x^3dx

c)

∫x4dx\int_{ }^{ }x^4dx

84.

∫(2x+1)5dx\int_{ }^{ }\left(2x+1\right)^5dx

a)

5(2x+1)45\left(2x+1\right)^4

b)

16(2x+1)6+C\frac{1}{6}\left(2x+1\right)^6+C

c)

15(2x+1)5+C\frac{1}{5}\left(2x+1\right)^5+C

d)

112(2x+1)6+C\frac{1}{12}\left(2x+1\right)^6+C

85.

∫e2xdx\int_{ }^{ }e^{2x}dx

a)

2e2x+C2e^{2x}+C

b)

2ex+C2e^x+C

c)

12e2x+C\frac{1}{2}e^{2x}+C

d)

12ex+C\frac{1}{2}e^x+C

86.

∫5xdx\int_{ }^{ }5^xdx

a)

x5ln⁡5+C\frac{x^5}{\ln5}+C

b)

5xln⁡5+C\frac{5^x}{\ln5}+C

c)

5xln⁡x+C\frac{5^x}{\ln x}+C

d)

5xln⁡x+C5^x\ln x+C

87.

∫cos⁡8xdx\int_{ }^{ }\cos8xdx

a)

sin⁡8x8+C\frac{\sin8x}{8}+C

b)

−sin⁡8x8+C-\frac{\sin8x}{8}+C

c)

8sin⁡8x+C8\sin8x+C

d)

−8sin⁡8x+C-8\sin8x+C

88.

∫e−xdx\int_{ }^{ }e^{-x}dx

a)

e−x+Ce^{-x}+C

b)

ex+Ce^x+C

c)

−e−x+C-e^{-x}+C

d)

−e−x-e^{-x}

89.

∫ 1xdx\int_{ }^{ }\ \frac{1}{x}dx

a)

ln⁡x+C\ln x+C

b)

−ln⁡x+C-\ln x+C

c)

ln⁡(x)+C\ln\left(x\right)+C

d)

ln⁡∣x∣+C\ln\left|x\right|+C

90.

∫ 45xdx\int_{ }^{ }\ \frac{4}{5x}dx

a)

4ln⁡∣5x∣+C4\ln\left|5x\right|+C

b)

5ln⁡∣5x∣+C5\ln\left|5x\right|+C

c)

45ln⁡∣5x∣+C\frac{4}{5}\ln\left|5x\right|+C

d)

54ln⁡∣5x∣+C\frac{5}{4}\ln\left|5x\right|+C

91.

∫  1cos⁡2xdx\int_{ }\ \ \frac{1}{\cos^2x}dx

a)

tan⁡x+C\tan x+C

b)

−tan⁡x+C-\tan x+C

c)

cot⁡x+C\cot x+C

d)

−cot⁡x+C-\cot x+C

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