5.6E: Mazoezi ya Sehemu ya 5.6

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Kwa mazoezi ya 1 - 8, compute kila muhimu kwa muda usiojulikana.

1)$\displaystyle ∫e^{2x}\,dx$

2)$\displaystyle ∫e^{−3x}\,dx$

Jibu: $\displaystyle ∫e^{−3x}\,dx \quad = \quad \frac{−1}{3}e^{−3x}+C$

3)$\displaystyle ∫2^x\,dx$

4)$\displaystyle ∫3^{−x}\,dx$

Jibu: $\displaystyle ∫3^{−x}\,dx \quad = \quad −\frac{3^{−x}}{\ln 3}+C$

5)$\displaystyle ∫\frac{1}{2x}\,dx$

6)$\displaystyle ∫\frac{2}{x}\,dx$

Jibu: $\displaystyle ∫\frac{2}{x}\,dx \quad = \quad 2\ln x+C \quad = \quad \ln(x^2)+C$

7)$\displaystyle ∫\frac{1}{x^2}\,dx$

8)$\displaystyle ∫\frac{1}{\sqrt{x}}\,dx$

Jibu: $\displaystyle ∫\frac{1}{\sqrt{x}}\,dx \quad = \quad 2\sqrt{x}+C$

Katika mazoezi 9 - 16, tafuta kila muhimu kwa kutumia mbadala zinazofaa.

9)$\displaystyle ∫\frac{\ln x}{x}\,dx$

10)$\displaystyle ∫\frac{dx}{x(\ln x)^2}$

Jibu: $\displaystyle ∫\frac{dx}{x(\ln x)^2} \quad = \quad −\frac{1}{\ln x}+C$

11)$\displaystyle ∫\frac{dx}{x\ln x}\quad (x>1)$

12)$\displaystyle ∫\frac{dx}{x\ln x\ln(\ln x)}$

Jibu: $\displaystyle ∫\frac{dx}{x\ln x\ln(\ln x)} \quad = \quad \ln(\ln(\ln x))+C$

13)$\displaystyle ∫\tan θ\,dθ$

14)$\displaystyle ∫\frac{\cos x−x\sin x}{x\cos x}\,dx$

Jibu: $\displaystyle ∫\frac{\cos x−x\sin x}{x\cos x}\,dx \quad = \quad \ln(x\cos x)+C$

15)$\displaystyle ∫\frac{\ln(\sin x)}{\tan x}\,dx$

16)$\displaystyle ∫\ln(\cos x)\tan x\,dx$

Jibu: $\displaystyle ∫\ln(\cos x)\tan x\,dx \quad = \quad −\dfrac{1}{2}(\ln(\cos(x)))^2+C$

17)$\displaystyle ∫xe^{−x^2}\,dx$

18)$\displaystyle ∫x^2e^{−x^3}\,dx$

Jibu: $\displaystyle ∫x^2e^{−x^3}\,dx \quad = \quad \dfrac{−e^{−x^3}}{3}+C$

19)$\displaystyle ∫e^{\sin x}\cos x\,dx$

20)$\displaystyle ∫e^{\tan x}\sec^2 x\,dx$

Jibu: $\displaystyle ∫e^{\tan x}\sec^2 x\,dx\quad = \quad e^{\tan x}+C$

21)$\displaystyle ∫\frac{e^{\ln x}}{x}\,dx $

22)$\displaystyle ∫\frac{e^{\ln(1−t)}}{1−t}\,dt$

Jibu: $\displaystyle ∫\frac{e^{\ln(1−t)}}{1−t}\,dt = \int \frac{1-t}{1-t}\,dt = \int 1\, dt \quad = \quad t+C$

Katika mazoezi 23 - 28, thibitisha kwa kutofautisha kwamba$\displaystyle ∫\ln x\,dx=x(\ln x−1)+C$, kisha utumie mabadiliko sahihi ya vigezo ili kukokotoa muhimu.

23)$\displaystyle ∫\ln x\,dx$ (Kidokezo:$\displaystyle ∫\ln x\,dx=\frac{1}{2}∫x\ln(x^2)\,dx$)

24)$\displaystyle ∫x^2\ln^2 x\,dx$

Jibu: $\displaystyle ∫x^2\ln^2 x\,dx \quad = \quad \dfrac{1}{9}x^3(\ln(x^3)−1)+C$

25)$\displaystyle ∫\frac{\ln x}{x^2}\,dx$ (Kidokezo: Weka$u=\dfrac{1}{x}.)$

26)$\displaystyle ∫\frac{\ln x}{\sqrt{x}}\,dx$ (Kidokezo: Weka$u=\sqrt{x}.)$

Jibu: $ \displaystyle ∫\frac{\ln x}{\sqrt{x}}\,dx \quad = \quad 2\sqrt{x}(\ln x−2)+C$

27) Andika muhimu kuelezea eneo chini ya grafu ya$y=\dfrac{1}{t}$ kutoka$ t=1$ kwa$e^x$ na kutathmini muhimu.

28) Andika muhimu kuelezea eneo chini ya grafu ya$y=e^t$ kati$t=0$ na$t=\ln x$, na kutathmini muhimu.

Jibu: $\displaystyle ∫^{\ln x}_0e^t\,dt=e^t\bigg|^{\ln x}_0=e^{\ln x}−e^0=x−1$

Katika mazoezi 29 - 35, tumia mbadala zinazofaa ili kuelezea integrals trigonometric kwa suala la nyimbo na logarithms.

29)$\displaystyle ∫\tan(2x)\,dx$

30)$\displaystyle ∫\frac{\sin(3x)−\cos(3x)}{\sin(3x)+\cos(3x)}\,dx$

Jibu: $ \displaystyle ∫\frac{\sin(3x)−\cos(3x)}{\sin(3x)+\cos(3x)}\,dx \quad = \quad −\frac{1}{3}\ln|\sin(3x)+\cos(3x)| + C$

31)$\displaystyle ∫\frac{x\sin(x^2)}{\cos(x^2)}\,dx$

32)$\displaystyle ∫x\csc(x^2)\,dx$

Jibu: $ \displaystyle ∫x\csc(x^2)\,dx \quad = \quad −\frac{1}{2}\ln∣\csc(x^2)+\cot(x^2)∣+C$

33)$\displaystyle ∫\ln(\cos x)\tan x\,dx$

34)$\displaystyle ∫\ln(\csc x)\cot x\,dx$

Jibu: $ \displaystyle ∫\ln(\csc x)\cot x\,dx \quad = \quad −\frac{1}{2}(\ln(\csc x))^2+C$

35)$\displaystyle ∫\frac{e^x−e^{−x}}{e^x+e^{−x}}\,dx$

Katika mazoezi 36 - 40, tathmini muhimu ya uhakika.

36)$\displaystyle ∫^2_1\frac{1+2x+x^2}{3x+3x^2+x^3}\,dx$

Jibu: $\displaystyle ∫^2_1\frac{1+2x+x^2}{3x+3x^2+x^3}\,dx \quad = \quad \frac{1}{3}\ln\left(\tfrac{26}{7}\right)$

37)$\displaystyle ∫^{π/4}_0\tan x\,dx$

38)$\displaystyle ∫^{π/3}_0\frac{\sin x−\cos x}{\sin x+\cos x}\,dx$

Jibu: $\displaystyle ∫^{π/3}_0\frac{\sin x−\cos x}{\sin x+\cos x}\,dx \quad = \quad \ln(\sqrt{3}−1)$

39)$\displaystyle ∫^{π/2}_{π/6}\csc x\,dx$

40)$\displaystyle ∫^{π/3}_{π/4}\cot x\,dx$

Jibu: $\displaystyle ∫^{π/3}_{π/4}\cot x\,dx \quad = \quad \frac{1}{2}\ln\frac{3}{2}$

Katika mazoezi 41 - 46, kuunganisha kwa kutumia ubadilishaji ulioonyeshwa.

41)$\displaystyle ∫\frac{x}{x−100}\,dx;\quad u=x−100$

42)$\displaystyle ∫\frac{y−1}{y+1}\,dy;\quad u=y+1$

Jibu: $ \displaystyle ∫\frac{y−1}{y+1}\,dy \quad = \quad y−2\ln|y+1|+C$

43)$\displaystyle ∫\frac{1−x^2}{3x−x^3}\,dx;\quad u=3x−x^3$

44)$\displaystyle ∫\frac{\sin x+\cos x}{\sin x−\cos x}\,dx;\quad u=\sin x−\cos x$

Jibu: $\displaystyle ∫\frac{\sin x+\cos x}{\sin x−\cos x}\,dx \quad=\quad \ln|\sin x−\cos x|+C$

45)$\displaystyle ∫e^{2x}\sqrt{1−e^{2x}}\,dx;\quad u=e^{2x}$

46)$\displaystyle ∫\ln(x)\frac{\sqrt{1−(\ln x)^2}}{x}\,dx;\quad u=\ln x$

Jibu: $\displaystyle ∫\ln(x)\frac{\sqrt{1−(\ln x)^2}}{x}\,dx \quad = \quad −\frac{1}{3}(1−(\ln x^2))^{3/2}+C$

47)$\displaystyle \int \frac{\sqrt{x}}{\sqrt{x} + 2}\,dx; \quad u = \sqrt{x} + 2$

Jibu: $\displaystyle \int \frac{\sqrt{x}}{\sqrt{x} + 2}\,dx \quad = \quad \left( \sqrt{x} + 2 \right)^2 - 8\left( \sqrt{x} + 2 \right) + 8\ln\left( \sqrt{x} + 2 \right) + C$

48)$\displaystyle \int e^x\sec(e^x+1)\tan(e^x+1)\,dx; \quad u = e^{x} + 1$

Jibu: $\displaystyle \int e^x\sec(e^x+1)\tan(e^x+1)\,dx \quad = \quad \sec(e^x+1) + C$

Katika mazoezi 49 - 54, sema kama makadirio ya mwisho ya haki-overestimates au underestimates eneo halisi. Kisha uhesabu makadirio ya mwisho ya mwisho$R_{50}$ na kutatua eneo halisi.

49) [T]$y=e^x$ juu$[0,1]$

50) [T]$ y=e^{−x}$ juu$[0,1]$

Jibu: Kwa kuwa$f$ inapungua, makadirio ya mwisho ya mwisho yanapunguza eneo hilo.
Suluhisho halisi:$\dfrac{e−1}{e},\quad R_{50}=0.6258$.

51) [T]$y=\ln(x)$ juu$[1,2]$

52) [T]$y=\dfrac{x+1}{x^2+2x+6}$ juu$ [0,1]$

Jibu: Kwa kuwa$f$ inaongezeka, makadirio ya mwisho ya mwisho yanazidi eneo hilo.
Suluhisho halisi:$\dfrac{2\ln(3)−\ln(6)}{2},\quad R_{50}=0.2033.$

53) [T]$y=2^x$ juu$[−1,0]$

54) [T]$ y=−2^{−x}$ juu$ [0,1]$

Jibu: Kwa kuwa$f$ inaongezeka, makadirio ya mwisho ya mwisho yanazidi eneo hilo (eneo halisi ni idadi kubwa hasi).
Suluhisho halisi:$−\dfrac{1}{\ln(4)},\quad R_{50}=−0.7164.$

Katika mazoezi 55 - 58,$f(x)≥0$ kwa$a≤x≤b$. Pata eneo chini ya grafu ya$f(x)$ kati ya maadili$b$ yaliyotolewa$a$ na kwa kuunganisha.

55)$f(x)=\dfrac{\log_{10}(x)}{x};\quad a=10,b=100$

56)$f(x)=\dfrac{\log_2(x)}{x};\quad a=32,b=64$

Jibu: $\dfrac{11}{2}\ln 2$

57)$f(x)=2^{−x};\quad a=1,b=2$

58)$f(x)=2^{−x};\quad a=3,b=4$

Jibu: $\dfrac{1}{\ln(65,536)}$

59) Pata eneo chini ya grafu ya kazi$ f(x)=xe^{−x^2}$ kati$x=0$ na$x=5$.

60) Compute muhimu ya$f(x)=xe^{−x^2}$ na kupata thamani ndogo ya$N$ vile kwamba eneo chini ya grafu$f(x)=xe^{−x^2}$ kati$ x=N$ na$x=N+10$ ni, saa zaidi,$0.01$.

Jibu: $\displaystyle ∫^{N+1}_Nxe^{−x^2}\,dx=\frac{1}{2}(e^{−N^2}−e^{−(N+1)^2}).$Kiasi ni chini ya$0.01$ wakati$N=2$.

61) Kupata kikomo, kama$N$ huelekea infinity, ya eneo chini ya grafu ya$f(x)=xe^{−x^2}$ kati$x=0$ na$x=5$.

62) Onyesha kwamba$\displaystyle ∫^b_a\frac{dt}{t}=∫^{1/a}_{1/b}\frac{dt}{t}$ wakati$0<a≤b$.

Jibu: $\displaystyle ∫^b_a\frac{dx}{x}=\ln(b)−\ln(a)=\ln(\frac{1}{a})−\ln(\frac{1}{b})=∫^{1/a}_{1/b}\frac{dx}{x}$

63) Tuseme kwamba$f(x)>0$ kwa wote$x$$f$ na kwamba na$g$ ni tofauti. Kutumia utambulisho$ f^g=e^{g\ln f}$ na utawala mnyororo kupata derivative ya$ f^g$.

64) Tumia zoezi la awali ili kupata antiderivative ya$h(x)=x^x(1+\ln x)$ na kutathmini$\displaystyle ∫^3_2x^x(1+\ln x)\,dx$.

Jibu: 23

65) Onyesha kwamba kama$c>0$, basi muhimu ya$\frac{1}{x}$ kutoka$ac$ kwa$bc$ $(\text{for}\,0<a<b)$ni sawa na muhimu ya$\frac{1}{x}$ kutoka$a$ kwa $b$.

Mazoezi yafuatayo yanalenga kupata mali ya msingi ya logi ya asili kuanzia ufafanuzi$\displaystyle \ln(x)=∫^x_1\frac{dt}{t}$, kwa kutumia mali ya muhimu ya uhakika na kufanya hakuna mawazo zaidi.

66) Tumia utambulisho$\displaystyle \ln(x)=∫^x_1\frac{dt}{t}$ kupata utambulisho$\ln\left(\dfrac{1}{x}\right)=−\ln x$.

Jibu: Tunaweza kudhani kwamba$x>1$, hivyo$\dfrac{1}{x}<1.$ Kisha,$\displaystyle ∫^{1/x}_{1}\frac{dt}{t}$. Sasa fanya ubadilishaji$u=\dfrac{1}{t}$, hivyo$du=−\dfrac{dt}{t^2}$ na$\dfrac{du}{u}=−\dfrac{dt}{t}$, na ubadilishe mwisho:$\displaystyle ∫^{1/x}_1\frac{dt}{t}=−∫^x_1\frac{du}{u}=−\ln x.$

67) Matumizi ya mabadiliko ya kutofautiana katika$\displaystyle ∫^{xy}_1\frac{1}{t}\,dt$ muhimu kuonyesha kwamba$\ln xy=\ln x+\ln y$ kwa$ x,y>0$.

68) Matumizi$\displaystyle \ln x=∫^x_1\frac{dt}{x}$ utambulisho kuonyesha kwamba$\ln(x)$ ni kuongeza kazi ya$x$ juu$[0,∞)$, na kutumia mazoezi ya awali kuonyesha kwamba mbalimbali ya$\ln(x)$ ni$(−∞,∞)$. Bila mawazo yoyote zaidi, kuhitimisha kwamba$\ln(x)$ ina kazi inverse defined juu ya$ (−∞,∞).$

69) Kujifanya, kwa muda, kwamba hatujui kwamba$e^x$ ni kazi inverse ya$\ln(x)$, lakini kukumbuka kwamba$\ln(x)$ ina kazi inverse defined juu ya$ (−∞,∞)$. Piga simu$E$. Tumia utambulisho$\ln xy=\ln x+\ln y$ wa kutambua kwamba$E(a+b)=E(a)E(b)$ kwa idadi yoyote halisi$a$,$b$.

70) Kujifanya, kwa muda, kwamba hatujui kwamba$ e^x$ ni kazi inverse ya$\ln x$, lakini kukumbuka kwamba$ \ln x$ ina kazi inverse defined juu ya$(−∞,∞)$. Piga simu$E$. Onyesha kwamba$E'(t)=E(t).$

Jibu: $x=E(\ln(x)).$Kisha,$1=\dfrac{E'(\ln x)}{x}$ au$x=E'(\ln x)$. Kwa kuwa idadi yoyote$t$ inaweza kuandikwa$t=\ln x$ kwa baadhi$x$, na kwa vile$t$ tuna$x=E(t)$, inafuata kwamba kwa yeyote$t,\,E'(t)=E(t).$

71) muhimu sine, hufafanuliwa kama$\displaystyle S(x)=∫^x_0\frac{\sin t}{t}\,dt$ ni kiasi muhimu katika uhandisi. Ingawa haina formula rahisi imefungwa, inawezekana kukadiria tabia yake kwa kubwa$x$. Onyesha kwamba kwa$k≥1,\quad |S(2πk)−S(2π(k+1))|≤\dfrac{1}{k(2k+1)π}.$ (kidokezo:$ \sin(t+π)=−\sin t$)

72) [T] Usambazaji wa kawaida katika uwezekano unatolewa na$p(x)=\dfrac{1}{σ\sqrt{2π}}e^{−(x−μ)^2/2σ^2}$, wapi$σ$ kupotoka kwa kiwango na$μ$ ni wastani. Usambazaji wa kawaida wa kawaida kwa uwezekano,$p_s$, unafanana$ μ=0$ na$σ=1$. Compute makadirio ya mwisho ya kushoto$R_{10}$ na$R_{100}$ ya$\displaystyle ∫^1_{−1}\frac{1}{\sqrt{2π}}e^{−x^{2/2}}\,dx.$

Jibu: $R_{10}=0.6811,\quad R_{100}=0.6827$

$Grafu ya kazi f (x) = .5 * (sqrt (2) *e^ (-.5x^ 2))/sqrt (pi). Ni chini ya ufunguzi Curve ambayo ni symmetric katika mhimili y, kuvuka saa kuhusu (0, .4). Inakaribia 0 kama x inakwenda infinity chanya na hasi. Kati ya 1 na -1, rectangles kumi hutolewa kwa makadirio ya mwisho ya eneo chini ya pembe.$

73) [T] Compute haki endpoint makadirio$R_{50}$ na$R_{100}$ ya$\displaystyle ∫^5_{−3}\frac{1}{2\sqrt{2π}}e^{−(x−1)^2/8}$.

Wachangiaji na Majina

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Paul Seeburger (Monroe Community College) added problems 47-48 to Section 5.6 exercises.