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7.2E: Mazoezi ya Sehemu ya 7.2

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  • Page ID
    178864

    • Edwin “Jed” Herman & Gilbert Strang
    • OpenStax
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    Jaza tupu ili kutoa taarifa ya kweli.

    1)\(\sin^2x+\) _______\( =1\)

    Jibu
    \(\cos^2x\)

    2)\(\sec^2x−1=\) _______

    Jibu
    \(\tan^2x\)

    Tumia utambulisho ili kupunguza nguvu ya kazi ya trigonometric kwa kazi ya trigonometric iliyofufuliwa kwa nguvu ya kwanza.

    3)\(\sin^2x=\) _______

    Jibu
    \(\dfrac{1−\cos(2x)}{2}\)

    4)\(\cos^2x=\) _______

    Jibu
    \(\dfrac{1+\cos(2x)}{2}\)

    Kutathmini kila moja ya integrals zifuatazo na\(u\) -badala.

    5)\(\displaystyle ∫\sin^3x\cos x\,dx\)

    Jibu
    \(\displaystyle ∫\sin^3x\cos x\,dx \quad = \quad \frac{\sin^4x}{4}+C\)

    6)\(\displaystyle ∫\sqrt{\cos x}\sin x\,dx\)

    7)\(\displaystyle ∫\tan^5(2x)\sec^2(2x)\,dx\)

    Jibu
    \(\displaystyle ∫\tan^5(2x)\sec^2(2x)\,dx \quad = \quad \tfrac{1}{12}\tan^6(2x)+C\)

    8)\(\displaystyle ∫\sin^7(2x)\cos(2x)\,dx\)

    9)\(\displaystyle ∫\tan(\frac{x}{2})\sec^2(\frac{x}{2})\,dx\)

    Jibu
    \(\displaystyle ∫\tan(\frac{x}{2})\sec^2(\frac{x}{2})\,dx \quad = \quad \tan^2(\frac{x}{2})+C\)

    10)\(\displaystyle ∫\tan^2x\sec^2x\,dx\)

    Futa integrals zifuatazo kwa kutumia miongozo ya kuunganisha nguvu za kazi za trigonometric. Tumia CAS ili uangalie ufumbuzi. (Kumbuka: Baadhi ya matatizo yanaweza kufanywa kwa kutumia mbinu za ushirikiano zilizojifunza hapo awali.)

    11)\(\displaystyle ∫\sin^3x\,dx\)

    Jibu
    \(\displaystyle ∫\sin^3x\,dx \quad = \quad −\frac{3\cos x}{4}+\tfrac{1}{12}\cos(3x)+C=−\cos x+\frac{\cos^3x}{3}+C\)

    12)\(\displaystyle ∫\cos^3x\,dx\)

    13)\(\displaystyle ∫\sin x\cos x\,dx\)

    Jibu
    \(\displaystyle ∫\sin x\cos x\,dx \quad = \quad −\tfrac{1}{2}\cos^2x+C\)

    14)\(\displaystyle ∫\cos^5x\,dx\)

    15)\(\displaystyle ∫\sin^5x\cos^2x\,dx\)

    Jibu
    \(\displaystyle ∫\sin^5x\cos^2x\,dx \quad = \quad −\frac{5\cos x}{64}−\tfrac{1}{192}\cos(3x)+\tfrac{3}{320}\cos(5x)−\tfrac{1}{448}\cos(7x)+C\)

    16)\(\displaystyle ∫\sin^3x\cos^3x\,dx\)

    17)\(\displaystyle ∫\sqrt{\sin x}\cos x\,dx\)

    Jibu
    \(\displaystyle ∫\sqrt{\sin x}\cos x\,dx \quad = \quad \tfrac{2}{3}(\sin x)^{3/2}+C\)

    18)\(\displaystyle ∫\sqrt{\sin x}\cos^3x\,dx\)

    19)\(\displaystyle ∫\sec x\tan x\,dx\)

    Jibu
    \(\displaystyle ∫\sec x\tan x\,dx \quad = \quad \sec x+C\)

    20)\(\displaystyle ∫\tan(5x)\,dx\)

    21)\(\displaystyle ∫\tan^2x\sec x\,dx\)

    Jibu
    \(\displaystyle ∫\tan^2x\sec x\,dx \quad = \quad \tfrac{1}{2}\sec x\tan x−\tfrac{1}{2}\ln(\sec x+\tan x)+C\)

    22)\(\displaystyle ∫\tan x\sec^3x\,dx\)

    23)\(\displaystyle ∫\sec^4x\,dx\)

    Jibu
    \(\displaystyle ∫\sec^4x\,dx \quad = \quad \frac{2\tan x}{3}+\tfrac{1}{3}\sec^2 x\tan x=\tan x+\frac{\tan^3x}{3}+C\)

    24)\(\displaystyle ∫\cot x\,dx\)

    25)\(\displaystyle ∫\csc x\,dx\)

    Jibu
    \(\displaystyle ∫\csc x\,dx \quad = \quad −\ln|\cot x+\csc x|+C\)

    26)\(\displaystyle ∫\frac{\tan^3x}{\sqrt{\sec x}}\,dx\)

    Kwa mazoezi 27 - 28, pata formula ya jumla kwa integrals.

    27)\(\displaystyle ∫\sin^2ax\cos ax\,dx\)

    Jibu
    \(\displaystyle ∫\sin^2ax\cos ax\,dx \quad = \quad \frac{\sin^3(ax)}{3a}+C\)

    28)\(\displaystyle ∫\sin ax\cos ax\,dx.\)

    Tumia formula mbili za angle ili kutathmini integrals katika mazoezi 29 - 34.

    29)\(\displaystyle ∫^π_0\sin^2x\,dx\)

    Jibu
    \(\displaystyle ∫^π_0\sin^2x\,dx \quad = \quad \frac{π}{2}\)

    30)\(\displaystyle ∫^π_0\sin^4 x\,dx\)

    31)\(\displaystyle ∫\cos^2 3x\,dx\)

    Jibu
    \(\displaystyle ∫\cos^2 3x\,dx \quad = \quad \frac{x}{2}+\tfrac{1}{12}\sin(6x)+C\)

    32)\(\displaystyle ∫\sin^2x\cos^2x\,dx\)

    33)\(\displaystyle ∫\sin^2x\,dx+∫\cos^2x\,dx\)

    Jibu
    \(\displaystyle ∫\sin^2x\,dx+∫\cos^2x\,dx \quad = \quad x+C\)

    34)\(\displaystyle ∫\sin^2 x\cos^2(2x)\,dx\)

    Kwa mazoezi 35 - 43, tathmini integrals uhakika. Express majibu katika fomu halisi wakati wowote iwezekanavyo.

    35)\(\displaystyle ∫^{2π}_0\cos x\sin 2x\,dx\)

    Jibu
    \(\displaystyle ∫^{2π}_0\cos x\sin 2x\,dx \quad = \quad 0\)

    36)\(\displaystyle ∫^π_0\sin 3x\sin 5x\,dx\)

    37)\(\displaystyle ∫^π_0\cos(99x)\sin(101x)\,dx\)

    Jibu
    \(\displaystyle ∫^π_0\cos(99x)\sin(101x)\,dx \quad = \quad 0\)

    38)\(\displaystyle ∫^π_{−π}\cos^2(3x)\,dx\)

    39)\(\displaystyle ∫^{2π}_0\sin x\sin(2x)\sin(3x)\,dx\)

    Jibu
    \(\displaystyle ∫^{2π}_0\sin x\sin(2x)\sin(3x)\,dx \quad = \quad 0\)

    40)\(\displaystyle ∫^{4π}_0\cos(x/2)\sin(x/2)\,dx\)

    41)\(\displaystyle ∫^{π/3}_{π/6}\frac{\cos^3x}{\sqrt{\sin x}}\,dx\) (Pindua jibu hili kwa maeneo matatu ya decimal.)

    Jibu
    \(\displaystyle ∫^{π/3}_{π/6}\frac{\cos^3x}{\sqrt{\sin x}}\,dx \quad \approx \quad 0.239\)

    42)\(\displaystyle ∫^{π/3}_{−π/3}\sqrt{\sec^2x−1}\,dx\)

    43)\(\displaystyle ∫^{π/2}_0\sqrt{1−\cos(2x)}\,dx\)

    Jibu
    \(\displaystyle ∫^{π/2}_0\sqrt{1−\cos(2x)}\,dx \quad = \quad \sqrt{2}\)

    44) Pata eneo la kanda lililofungwa na grafu za equations\(y=\sin x,\, y=\sin^3x,\, x=0,\) na\(x=\frac{π}{2}.\)

    45) Pata eneo la kanda lililofungwa na grafu za equations\(y=\cos^2x,\, y=\sin^2x,\, x=−\frac{π}{4},\) na\(x=\frac{π}{4}.\)

    Jibu
    \(A = 1 \,\text{unit}^2\)

    46) chembe hatua katika mstari wa moja kwa moja na kasi kazi\(v(t)=\sin(ωt)\cos^2(ωt).\) Kupata nafasi yake kazi\(x=f(t)\) kama\( f(0)=0.\)

    47) Pata thamani ya wastani ya kazi\(f(x)=\sin^2x\cos^3x\) zaidi ya muda\([−π,π].\)

    Jibu
    \(0\)

    Kwa mazoezi 48 - 49, tatua usawa tofauti.

    48)\(\dfrac{dy}{\,dx}=\sin^2x.\) Curve hupita kupitia hatua\((0,0).\)

    49)\(\dfrac{dy}{dθ}=\sin^4(πθ)\)

    Jibu
    \(f(x) = \dfrac{3θ}{8}−\tfrac{1}{4π}\sin(2πθ)+\tfrac{1}{32π}\sin(4πθ)+C\)

    50) Pata urefu wa pembe\(y=\ln(\csc x),\, \text{for}\,\tfrac{π}{4}≤x≤\tfrac{π}{2}.\)

    51) Pata urefu wa pembe\(y=\ln(\sin x),\, \text{for}\,\tfrac{π}{3}≤x≤\tfrac{π}{2}.\)

    Jibu
    \(s = \ln(\sqrt{3})\)

    52) Pata kiasi kilichozalishwa na kuzunguka pembe\(y=\cos(3x)\) kuhusu\(x\) -axis, kwa\( 0≤x≤\tfrac{π}{36}.\)

    Kwa mazoezi 53 - 54, tumia habari hii: Bidhaa ya ndani ya kazi mbili\(f\) na\(g\) zaidi\([a,b]\) inaelezwa na kazi\(\displaystyle f(x)⋅g(x)=⟨f,g⟩=∫^b_af⋅g\,dx.\) mbili tofauti\(f\) na\(g\) inasemekana kuwa orthogonal kama\(⟨f,g⟩=0.\)

    53) Onyesha kwamba\({\sin(2x),\, \cos(3x)}\) ni orthogonal juu ya muda\([−π,\, π]\).

    Jibu
    \(\displaystyle ∫^π_{−π}\sin(2x)\cos(3x)\,dx=0\)

    54) Tathmini\(\displaystyle ∫^π_{−π}\sin(mx)\cos(nx)\,dx.\)

    55) Unganisha\(y′=\sqrt{\tan x}\sec^4x.\)

    Jibu
    \(\displaystyle y = \int \sqrt{\tan x}\sec^4x \, dx \quad = \quad \tfrac{2}{3}\left(\tan x\right)^{3/2} + \tfrac{2}{7}\left(\tan x\right)^{7/2}+C= \tfrac{2}{21}\left(\tan x\right)^{3/2}\left[ 7 + 3\tan^2 x \right]+C\)

    Kwa kila jozi ya integrals katika mazoezi 56 - 57, kuamua ambayo ni vigumu zaidi kutathmini. Eleza hoja zako.

    56)\(\displaystyle ∫\sin^{456}x\cos x\,dx\) au\(\displaystyle ∫\sin^2x\cos^2x\,dx\)

    57)\(\displaystyle ∫\tan^{350}x\sec^2x\,dx\) au\(\displaystyle ∫\tan^{350}x\sec x\,dx\)

    Jibu
    Muhimu wa pili ni ngumu zaidi kwa sababu muhimu ya kwanza ni aina ya\(u\) kubadilisha tu.