7.7E: Mazoezi ya Sehemu ya 7.7

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Page ID: 178852

Edwin “Jed” Herman & Gilbert Strang
OpenStax

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Katika mazoezi ya 1 - 8, tathmini integrals zifuatazo. Ikiwa muhimu haipatikani, jibu “Inapungua.”

1)\(\displaystyle ∫^4_2\frac{dx}{(x−3)^2}\)

Jibu: Inapingana.

2)\(\displaystyle ∫^∞_0\frac{1}{4+x^2}\,dx\)

3)\(\displaystyle ∫^2_0\frac{1}{\sqrt{4−x^2}}\,dx\)

Jibu: Inajiunga kwa\(\frac{π}{2}\)

4)\(\displaystyle ∫^∞_1\frac{1}{x\ln x}\,dx\)

5)\(\displaystyle ∫^∞_1xe^{−x}\,dx\)

Jibu: Inajiunga kwa\(\frac{2}{e}\)

6)\(\displaystyle ∫^∞_{−∞}\frac{x}{x^2+1}\,dx\)

7) Bila kuunganisha, kuamua kama muhimu\(\displaystyle ∫^∞_1\frac{1}{\sqrt{x^3+1}}\,dx\) hujiunga au hutofautiana kwa kulinganisha kazi\(f(x)=\dfrac{1}{\sqrt{x^3+1}}\) na\(g(x)=\dfrac{1}{\sqrt{x^3}}\).

Jibu: Inajiunga.

8) Bila kuunganisha, tambua ikiwa ni muhimu\(\displaystyle ∫^∞_1\frac{1}{\sqrt{x+1}}\,dx\) hujiunga au hupungua.

Katika mazoezi 9 - 25, onyesha kama integrals zisizofaa hujiunga au kutofautiana. Ikiwezekana, tambua thamani ya viungo vinavyojiunga.

9)\(\displaystyle ∫^∞_0e^{−x}\cos x\,dx\)

Jibu: Inajiunga na\(\frac{1}{2}\).

10)\(\displaystyle ∫^∞_1\frac{\ln x}{x}\,dx\)

11)\(\displaystyle ∫^1_0\frac{\ln x}{\sqrt{x}}\,dx\)

Jibu: Inajiunga na\(-4\).

12)\(\displaystyle ∫^1_0\ln x\,dx\)

13)\(\displaystyle ∫^∞_{−∞}\frac{1}{x^2+1}\,dx\)

Jibu: Inajiunga na\(π\).

14)\(\displaystyle ∫^5_1\frac{dx}{\sqrt{x−1}}\)

15)\(\displaystyle ∫^2_{−2}\frac{dx}{(1+x)^2}\)

Jibu: Inapingana.

16)\(\displaystyle ∫^∞_0e^{−x}\,dx\)

17)\(\displaystyle ∫^∞_0\sin x\,dx\)

Jibu: Inapingana.

18)\(\displaystyle ∫^∞_{−∞}\frac{e^x}{1+e^{2x}}\,dx\)

19)\(\displaystyle ∫^1_0\frac{dx}{\sqrt[3]{x}}\)

Jibu: Inajiunga na\(1.5\).

20)\(\displaystyle ∫^2_0\frac{dx}{x^3}\)

21)\(\displaystyle ∫^2_{−1}\frac{dx}{x^3}\)

Jibu: Inapingana.

22)\(\displaystyle ∫^1_0\frac{dx}{\sqrt{1−x^2}}\)

23)\(\displaystyle ∫^3_0\frac{1}{x−1}\,dx\)

Jibu: Inapingana.

24)\(\displaystyle ∫^∞_1\frac{5}{x^3}\,dx\)

25)\(\displaystyle ∫^5_3\frac{5}{(x−4)^2}\,dx\)

Jibu: Inapingana.

Katika mazoezi ya 26 na 27, tambua ushirikiano wa kila moja ya vipengele vifuatavyo kwa kulinganisha na muhimu iliyotolewa. Ikiwa muhimu hujiunga, pata nambari ambayo hujiunga.

26)\(\displaystyle ∫^∞_1\frac{dx}{x^2+4x};\) kulinganisha na\(\displaystyle ∫^∞_1\frac{dx}{x^2}\).

27)\(\displaystyle ∫^∞_1\frac{dx}{\sqrt{x}+1};\) kulinganisha na\(\displaystyle ∫^∞_1\frac{dx}{2\sqrt{x}}\).

Jibu: Wote integrals wanatofautiana.

Katika mazoezi 28 - 38, tathmini integrals. Ikiwa muhimu hutofautiana, jibu “Inapungua.”

28)\(\displaystyle ∫^∞_1\frac{dx}{x^e}\)

29)\(\displaystyle ∫^1_0\frac{dx}{x^π}\)

Jibu: Inapingana.

30)\(\displaystyle ∫^1_0\frac{dx}{\sqrt{1−x}}\)

31)\(\displaystyle ∫^1_0\frac{dx}{1−x}\)

Jibu: Inapingana.

32)\(\displaystyle ∫^0_{−∞}\frac{dx}{x^2+1}\)

33)\(\displaystyle ∫^1_{−1}\frac{dx}{\sqrt{1−x^2}}\)

Jibu: Inajiunga na\(π\).

34)\(\displaystyle ∫^1_0\frac{\ln x}{x}\,dx\)

35)\(\displaystyle ∫^e_0\ln(x)\,dx\)

Jibu: Inajiunga na\(0\).

36)\(\displaystyle ∫^∞_0xe^{−x}\,dx\)

37)\(\displaystyle ∫^∞_{−∞}\frac{x}{(x^2+1)^2}\,dx\)

Jibu: Inajiunga na\(0\).

38)\(\displaystyle ∫^∞_0e^{−x}\,dx\)

Katika mazoezi 39 - 44, tathmini integrals zisizofaa. Kila moja ya vipengele hivi ina upungufu usio na mwisho ama mwisho au katika hatua ya ndani ya muda.

39)\(\displaystyle ∫^9_0\frac{dx}{\sqrt{9−x}}\)

Jibu: Inajiunga na\(6\).

40)\(\displaystyle ∫^1_{−27}\frac{dx}{x^{2/3}}\)

41)\(\displaystyle ∫^3_0\frac{dx}{\sqrt{9−x^2}}\)

Jibu: Inajiunga na\(\frac{π}{2}\).

42)\(\displaystyle ∫^{24}_6\frac{dt}{t\sqrt{t^2−36}}\)

43)\(\displaystyle ∫^4_0x\ln(4x)\,dx\)

Jibu: Inajiunga na\(8\ln(16)−4\).

44)\(\displaystyle ∫^3_0\frac{x}{\sqrt{9−x^2}}\,dx\)

45) Tathmini\(\displaystyle ∫^t_{.5}\frac{dx}{\sqrt{1−x^2}}.\) (Kuwa makini!) (Eleza jibu lako kwa kutumia sehemu tatu za decimal.)

Jibu: Inajiunga na kuhusu\(1.047\).

46) Tathmini\(\displaystyle ∫^4_1\frac{dx}{\sqrt{x^2−1}}.\) (Eleza jibu kwa fomu halisi.)

47) Tathmini\(\displaystyle ∫^∞_2\frac{dx}{(x^2−1)^{3/2}}.\)

Jibu: Inajiunga na\(−1+\frac{2}{\sqrt{3}}\).

48) Pata eneo la kanda katika quadrant ya kwanza kati ya pembe\(y=e^{−6x}\) na\(x\) -axis.

49) Pata eneo la kanda lililofungwa na pembe\(y=\dfrac{7}{x^2},\) ya\(x\) mhimili, na upande wa kushoto\(x=1.\)

Jibu: \(A = 7.0\)vitengo. ²

50) Kupata eneo chini ya Curve\(y=\dfrac{1}{(x+1)^{3/2}},\) imepakana upande wa kushoto na\(x=3.\)

51) Pata eneo chini ya\(y=\dfrac{5}{1+x^2}\) roboduara ya kwanza.

Jibu: \(A = \dfrac{5π}{2}\)vitengo. ²

52) Kupata kiasi cha imara yanayotokana na yanazunguka kuhusu\(x\) -axis kanda chini ya Curve\(y=\dfrac{3}{x}\) kutoka\(x=1\) kwa\(x=∞.\)

53) Kupata kiasi cha imara yanayotokana na yanazunguka juu ya\(y\) -axis kanda chini ya Curve\(y=6e^{−2x}\) katika roboduara ya kwanza.

Jibu: \(V = 3π\,\text{units}^3\)

54) Pata kiasi cha imara kilichozalishwa na kinachozunguka kuhusu\(x\) -axis eneo chini ya pembe\(y=3e^{−x}\) katika quadrant ya kwanza.

Kubadilisha Laplace ya kazi inayoendelea juu ya muda\([0,∞)\) hufafanuliwa na\(\displaystyle F(s)=∫^∞_0e^{−sx}f(x)\,dx\) (angalia Mradi wa Mwanafunzi). Ufafanuzi huu hutumiwa kutatua matatizo muhimu ya thamani ya awali katika usawa tofauti, kama ilivyojadiliwa baadaye. Domain ya\(F\) ni seti ya namba zote halisi s kama kwamba yasiyofaa muhimu hujiunga. Kupata Laplace kubadilisha\(F\) ya kila moja ya kazi zifuatazo na kutoa uwanja wa\(F\).

55)\(f(x)=1\)

Jibu: \(\dfrac{1}{s},\quad s>0\)

56)\(f(x)=x\)

57)\(f(x)=\cos(2x)\)

Jibu: \(\dfrac{s}{s^2+4},\quad s>0\)

58)\(f(x)=e^{ax}\)

59) Tumia formula kwa urefu wa arc ili kuonyesha kwamba mzunguko wa mduara\(x^2+y^2=1\) ni\(2π\).

Jibu: Majibu yatatofautiana.

kazi ni uwezekano wiani kazi kama satisfies ufafanuzi zifuatazo:\(\displaystyle ∫^∞_{−∞}f(t)\,dt=1\). uwezekano kwamba variable random\(x\) uongo kati ya a na b ni iliyotolewa na\(\displaystyle P(a≤x≤b)=∫^b_af(t)\,dt.\)

60) Onyesha kwamba\(\displaystyle f(x)=\begin{cases}0,&\text{if}\,x<0\\7e^{−7x},&\text{if}\,x≥0\end{cases}\) ni uwezekano wiani kazi.

61) Kupata uwezekano kwamba\(x\) ni kati\(0\) na\(0.3\). (Tumia kazi iliyoelezwa katika tatizo lililotangulia.) Tumia usahihi wa decimal wa mahali nne.

Jibu: 0.8775