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7.5E: Mazoezi ya Sehemu ya 7.5

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  • Page ID
    178844

    • Edwin “Jed” Herman & Gilbert Strang
    • OpenStax
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    Tumia meza ya integrals kutathmini integrals zifuatazo.

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

    2)\(\displaystyle ∫\frac{x+3}{x^2+2x+2}\,dx\)

    Jibu
    \(\displaystyle ∫\frac{x+3}{x^2+2x+2}\,dx = \tfrac{1}{2}\ln |x^2+2x+2|+2\arctan(x+1)+C\)

    3)\(\displaystyle ∫x^3\sqrt{1+2x^2}\,dx\)

    4)\(\displaystyle ∫\frac{1}{\sqrt{x^2+6x}}\,dx\)

    Jibu
    \(\displaystyle ∫\frac{1}{\sqrt{x^2+6x}}\,dx = \cosh^{−1}\left(\frac{x+3}{3}\right)+C\)

    5)\(\displaystyle ∫\frac{x}{x+1}\,dx\)

    6)\(\displaystyle ∫x⋅2^{x^2}\,dx\)

    Jibu
    \(\displaystyle ∫x⋅2^{x^2}\,dx = \frac{2^{x^2−1}}{\ln 2}+C\)

    7)\(\displaystyle ∫\frac{1}{4x^2+25}\,dx\)

    8)\(\displaystyle ∫\frac{dy}{\sqrt{4−y^2}}\)

    Jibu
    \(\displaystyle ∫\frac{dy}{\sqrt{4−y^2}} = \arcsin\left(\frac{y}{2}\right)+C\)

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

    10)\(\displaystyle ∫\csc(2w)\cot(2w)\,dw\)

    Jibu
    \(\displaystyle ∫\csc(2w)\cot(2w)\,dw = −\tfrac{1}{2}\csc(2w)+C\)

    11)\(\displaystyle ∫2^y\,dy\)

    12)\(\displaystyle ∫^1_0\frac{3x}{\sqrt{x^2+8}}\,dx\)

    Jibu
    \(\displaystyle ∫^1_0\frac{3x}{\sqrt{x^2+8}}\,dx = 9−6\sqrt{2}\)

    13)\(\displaystyle ∫^{1/4}_{−1/4}\sec^2(πx)\tan(πx)\,dx\)

    14)\(\displaystyle ∫^{π/2}_0\tan^2\left(\frac{x}{2}\right)\,dx\)

    Jibu
    \(\displaystyle ∫^{π/2}_0\tan^2\left(\frac{x}{2}\right)\,dx = 2−\frac{π}{2}\)

    15)\(\displaystyle ∫\cos^3x\,dx\)

    16)\(\displaystyle ∫\tan^5(3x)\,dx\)

    Jibu
    \(\displaystyle ∫\tan^5(3x)\,dx = \tfrac{1}{12}\tan^4(3x)−\tfrac{1}{6}\tan^2(3x)+\tfrac{1}{3}\ln|\sec 3x|+C\)

    17)\(\displaystyle ∫\sin^2y\cos^3y\,dy\)

    Tumia CAS kutathmini integrals zifuatazo. Majedwali pia yanaweza kutumika kuthibitisha majibu.

    18) [T]\(\displaystyle ∫\frac{dw}{1+\sec\left(\frac{w}{2}\right)}\)

    Jibu
    \(\displaystyle ∫\frac{dw}{1+\sec\left(\frac{w}{2}\right)} = 2\cot\left(\tfrac{w}{2}\right)−2\csc\left(\tfrac{w}{2}\right)+w+C\)

    19) [T]\(\displaystyle ∫\frac{dw}{1−\cos(7w)}\)

    20) [T]\(\displaystyle ∫^t_0\frac{dt}{4\cos t+3\sin t}\)

    Jibu
    \(\displaystyle ∫^t_0\frac{dt}{4\cos t+3\sin t} = \tfrac{1}{5}\ln\Big|\frac{2(5+4\sin t−3\cos t)}{4\cos t+3\sin t}\Big|\)

    21) [T]\(\displaystyle ∫\frac{\sqrt{x^2−9}}{3x}\,dx\)

    22) [T]\(\displaystyle ∫\frac{dx}{x^{1/2}+x^{1/3}}\)

    Jibu
    \(\displaystyle ∫\frac{dx}{x^{1/2}+x^{1/3}} = 6x^{1/6}−3x^{1/3}+2\sqrt{x}−6\ln[1+x^{1/6}]+C\)

    23) [T]\(\displaystyle ∫\frac{dx}{x\sqrt{x−1}}\)

    24) [T]\(\displaystyle ∫x^3\sin x\,dx\)

    Jibu
    \(\displaystyle ∫x^3\sin x\,dx = −x^3\cos x+3x^2\sin x+6x\cos x−6\sin x+C\)

    25) [T]\(\displaystyle ∫x\sqrt{x^4−9}\,dx\)

    26) [T]\(\displaystyle ∫\frac{x}{1+e^{−x^2}}\,dx\)

    Jibu
    \(\displaystyle ∫\frac{x}{1+e^{−x^2}}\,dx = \tfrac{1}{2}\left(x^2+\ln|1+e^{−x^2}|\right)+C\)

    27) [T]\(\displaystyle ∫\frac{\sqrt{3−5x}}{2x}\,dx\)

    28) [T]\(\displaystyle ∫\frac{dx}{x\sqrt{x−1}}\)

    Jibu
    \(\displaystyle ∫\frac{dx}{x\sqrt{x−1}} = 2\arctan\big(\sqrt{x−1}\big)+C\)

    29) [T]\(\displaystyle ∫e^x\cos^{−1}(e^x)\,dx\)

    Tumia calculator au CAS kutathmini integrals zifuatazo.

    30) [T]\(\displaystyle ∫^{π/4}_0\cos 2x \, dx\)

    Jibu
    \(\displaystyle ∫^{π/4}_0\cos 2x \, dx = 0.5=\frac{1}{2}\)

    31) [T]\(\displaystyle ∫^1_0x⋅e^{−x^2}\,dx\)

    32) [T]\(\displaystyle ∫^8_0\frac{2x}{\sqrt{x^2+36}}\,dx\)

    Jibu
    \(\displaystyle ∫^8_0\frac{2x}{\sqrt{x^2+36}}\,dx = 8.0\)

    33) [T]\(\displaystyle ∫^{2/\sqrt{3}}_0\frac{1}{4+9x^2}\,dx\)

    34) [T]\(\displaystyle ∫\frac{dx}{x^2+4x+13}\)

    Jibu
    \(\displaystyle ∫\frac{dx}{x^2+4x+13} = \tfrac{1}{3}\arctan\left(\tfrac{1}{3}(x+2)\right)+C\)

    35) [T]\(\displaystyle ∫\frac{dx}{1+\sin x}\)

    Tumia meza ili kutathmini integrals. Unaweza haja ya kukamilisha mraba au kubadilisha vigezo kuweka muhimu katika fomu iliyotolewa katika meza.

    36)\(\displaystyle ∫\frac{dx}{x^2+2x+10}\)

    Jibu
    \(\displaystyle ∫\frac{dx}{x^2+2x+10} = \tfrac{1}{3}\arctan\left(\frac{x+1}{3}\right)+C\)

    37)\(\displaystyle ∫\frac{dx}{\sqrt{x^2−6x}}\)

    38)\(\displaystyle ∫\frac{e^x}{\sqrt{e^{2x}−4}}\,dx\)

    Jibu
    \(\displaystyle ∫\frac{e^x}{\sqrt{e^{2x}−4}}\,dx = \ln\left(e^x+\sqrt{4+e^{2x}}\right)+C\)

    39)\(\displaystyle ∫\frac{\cos x}{\sin^2x+2\sin x}\,dx\)

    40)\(\displaystyle ∫\frac{\arctan(x^3)}{x^4}\,dx\)

    Jibu
    \(\displaystyle ∫\frac{\arctan(x^3)}{x^4}\,dx = \ln x−\tfrac{1}{6}\ln(x^6+1)−\frac{\arctan(x^3)}{3x^3}+C\)

    41)\(\displaystyle ∫\frac{\ln|x|\arcsin\left(\ln|x|\right)}{x}\,dx\)

    Tumia meza kufanya ushirikiano.

    42)\(\displaystyle ∫\frac{dx}{\sqrt{x^2+16}}\)

    Jibu
    \(\displaystyle ∫\frac{dx}{\sqrt{x^2+16}} = \ln |x|+\sqrt{16+x^2}∣+C\)

    43)\(\displaystyle ∫\frac{3x}{2x+7}\,dx\)

    44)\(\displaystyle ∫\frac{dx}{1−\cos 4x}\)

    Jibu
    \(\displaystyle ∫\frac{dx}{1−\cos 4x} = −\frac{1}{4}\cot 2x+C\)

    45)\(\displaystyle ∫\frac{dx}{\sqrt{4x+1}}\)

    46) Kupata eneo imepakana\(y(4+25x^2)=5,\;x=0,\;y=0,\) na na\(x=4.\) Matumizi meza ya integrals au CAS.

    Jibu
    \(\frac{1}{2}\arctan 10\)vitengo ²

    47) Eneo lililofungwa kati ya pembe\(y=\dfrac{1}{\sqrt{1+\cos x}}, \; 0.3≤x≤1.1,\) na\(x\) -axis linahusu\(x\) -axis kuzalisha imara. Tumia meza ya integrals kupata kiasi cha imara yanayotokana. (Pindua jibu kwa maeneo mawili ya decimal.)

    48) Tumia mbadala na meza ya integrals ili kupata eneo la uso unaozalishwa na kuzunguka curve\(y=e^x,\; 0≤x≤3,\) kuhusu\(x\) -axis. (Pindua jibu kwa maeneo mawili ya decimal.)

    Jibu
    \(1276.14\)vitengo ²

    49) [T] Matumizi meza muhimu na calculator kupata eneo la uso yanayotokana na yanazunguka Curve\(y=\dfrac{x^2}{2},\; 0≤x≤1,\) kuhusu\(x\) -axis. (Pindua jibu kwa maeneo mawili ya decimal.)

    50) [T] Matumizi CAS au meza ili kupata eneo la uso yanayotokana na yanazunguka Curve\(y=\cos x,\; 0≤x≤\frac{π}{2},\) kuhusu\(x\) -axis. (Pindua jibu kwa maeneo mawili ya decimal.)

    Jibu
    \(7.21\)vitengo ²

    51) Kupata urefu wa Curve\(y=\dfrac{x^2}{4}\) juu\([0,8]\).

    52) Pata urefu wa pembe\(y=e^x\) juu\([0,\,\ln(2)].\)

    Jibu
    \(\left(\sqrt{5}−\sqrt{2}+\ln\Big|\frac{2+2\sqrt{2}}{1+\sqrt{5}}\Big|\right)\)vitengo

    53) Pata eneo la uso lililoundwa na kuzunguka grafu ya\(y=2\sqrt{x}\) juu ya muda\([0,9]\) kuhusu\(x\) -axis.

    54) Pata thamani ya wastani ya kazi\(f(x)=\dfrac{1}{x^2+1}\) zaidi ya muda\([−3,3].\)

    Jibu
    \(\frac{1}{3}\arctan(3)≈0.416\)

    55) Takriban urefu wa arc wa curve\(y=\tan πx\) juu ya muda\(\left[0,\frac{1}{4}\right]\). (Pindua jibu kwa maeneo matatu ya decimal.)