積分習題及詳解14

02-07

479.

$eqalign{ & {{ ext{y}}^2} + 3x = 1, cr & frac{{{ ext{d}}left( {{{ ext{y}}^2}} ight)}}{{{ ext{d}}x}} = frac{{{ ext{d}}left( {{{ ext{y}}^2}} ight)}}{{{ ext{d}}y}} cdot frac{{{ ext{dy}}}}{{{ ext{d}}x}} = 2y cdot frac{{{ ext{dy}}}}{{{ ext{d}}x}} = 2yy cr & 2yy + 3 = 0 cr}$

441.

$[egin{gathered} int {frac{{1 + sqrt {1 - {x^2}} }}{{1 - sqrt {1 - {x^2}} }}} {kern 1pt} { ext{d}}x = - int {frac{{sqrt {1 - {x^2}} + 1}}{{sqrt {1 - {x^2}} - 1}}{kern 1pt} { ext{d}}x} hfill \ sqrt {1 - {x^2}} + 1 = sqrt {1 - {x^2}} - 1 + 2, hfill \ = - left( {2int {frac{1}{{sqrt {1 - {x^2}} - 1}}{kern 1pt} { ext{d}}x} + int {1{kern 1pt} { ext{d}}x} } ight) hfill \ = 2int {frac{{sqrt {1 - {x^2}} + 1}}{{{x^2}}}{kern 1pt} { ext{d}}x} - x hfill \ = 2left( {int {frac{{sqrt {1 - {x^2}} }}{{{x^2}}}{kern 1pt} { ext{d}}x} + int {frac{1}{{{x^2}}}{kern 1pt} { ext{d}}x} } ight) - x hfill \ u = sqrt {1 - {x^2}} ,v = frac{1}{{{x^2}}}, hfill \ u = - frac{x}{{sqrt {1 - {x^2}} }},v = - frac{1}{x}, hfill \ end{gathered} ]$

$[egin{gathered} = 2left( { - int {frac{1}{{sqrt {1 - {x^2}} }}{kern 1pt} { ext{d}}x} - frac{{sqrt {1 - {x^2}} }}{x} - frac{1}{x}} ight) - x hfill \ = 2left( { - arcsin x - frac{{sqrt {1 - {x^2}} }}{x} - frac{1}{x}} ight) - x hfill \ = - 2arcsin x - frac{{2sqrt {1 - {x^2}} }}{x} - frac{2}{x} - x hfill \ end{gathered} ]$

442.

$[egin{gathered} int {frac{1}{{xsqrt {{x^2} - 1} }}{kern 1pt} { ext{d}}x} hfill \ u = sqrt {{x^2} - 1} ,frac{{{ ext{d}}u}}{{{ ext{d}}x}} = frac{x}{{sqrt {{x^2} - 1} }}, hfill \ = int {frac{1}{{{u^2} + 1}}{kern 1pt} { ext{d}}u} = arctan sqrt {{x^2} - 1} + C hfill \ = - arcsin frac{1}{{left| x ight|}} + C hfill \ end{gathered} ]$

443.

$[egin{gathered} int {frac{1}{{{{left( {1 + {x^2}} ight)}^{frac{3}{2}}}}}} {kern 1pt} { ext{d}}x hfill \ x = an u,u = arctan x,frac{{{ ext{d}}x}}{{{ ext{d}}u}} = {sec ^2}u, hfill \ = int {frac{{{{sec }^2}u}}{{{{left( {{{ an }^2}u + 1} ight)}^{frac{3}{2}}}}}{kern 1pt} { ext{d}}u} hfill \ = int {frac{1}{{sec u}}{kern 1pt} { ext{d}}u} = int {cos u{kern 1pt} { ext{d}}u} = sin u hfill \ = sin left( {arctan x} ight) = frac{x}{{sqrt {{x^2} + 1} }} hfill \ = frac{x}{{sqrt {{x^2} + 1} }} + C hfill \ end{gathered} ]$

444.

$[frac{2}{{{t^2} - 1}} = 2frac{1}{{{t^2} - 1}} = 2(frac{A}{{t + 1}} + frac{B}{{t + 1}})]$

445.

$[egin{gathered} int {frac{{ an x}}{{{{cos }^2}x}}{kern 1pt} { ext{d}}x} = int {frac{{sin x}}{{{{cos }^3}x}}{kern 1pt} { ext{d}}x} hfill \ u = cos x,frac{{{ ext{d}}u}}{{{ ext{d}}x}} = - sin x, hfill \ = - int {frac{1}{{{u^3}}}{kern 1pt} { ext{d}}u} = frac{1}{{2{u^2}}} hfill \ = frac{1}{{2{{cos }^2}x}} + C hfill \ end{gathered} ]$

446.

$[egin{gathered} int {left( {x - {x^3}} ight){{ ext{e}}^{{x^2}}}{kern 1pt} { ext{d}}x} = - int {xleft( {{x^2} - 1} ight){{ ext{e}}^{{x^2}}}{kern 1pt} { ext{d}}x} hfill \ u = {x^2}, hfill \ = - frac{1}{2}int {left( {u - 1} ight){{ ext{e}}^u}{kern 1pt} { ext{d}}u} hfill \ f = { ext{u}} - { ext{1,g}} = {{ ext{e}}^u}, hfill \ f = 1,g = {{ ext{e}}^u}, hfill \ = - frac{1}{2}left[ {left( {u - 1} ight){{ ext{e}}^u} - int {{{ ext{e}}^u}{kern 1pt} { ext{d}}u} } ight] hfill \ = - frac{1}{2}left[ {left( {u - 1} ight){{ ext{e}}^u} - int {{{ ext{e}}^u}{kern 1pt} { ext{d}}u} } ight] hfill \ = - frac{{left( {u - 1} ight){{ ext{e}}^u}}}{2} + frac{{{{ ext{e}}^u}}}{2} hfill \ = frac{{{{ ext{e}}^{{x^2}}}}}{2} - frac{{left( {{x^2} - 1} ight){{ ext{e}}^{{x^2}}}}}{2} + C hfill \ = - frac{{left( {{x^2} - 2} ight){{ ext{e}}^{{x^2}}}}}{2} + C hfill \ end{gathered} ]$

$[intlimits_0^1 {left( {x - {x^3}} ight)} {{ ext{e}}^{{x^2}}}{kern 1pt} { ext{d}}x = frac{{ ext{e}}}{2} - 1]$

447.

$[egin{gathered} int {frac{{cos 2x}}{{{{cos }^2}x{{sin }^2}x}}{kern 1pt} { ext{d}}x} hfill \ = int {4cos 2x{{csc }^2}2x{kern 1pt} { ext{d}}x} hfill \ u = sin 2x,frac{{{ ext{d}}u}}{{{ ext{d}}x}} = 2cos 2x, hfill \ = 2int {frac{1}{{{u^2}}}{kern 1pt} { ext{d}}u} = - frac{2}{u} hfill \ = - frac{2}{{sin 2x}} + C hfill \ = - 2csc 2x + C hfill \ = - an x - frac{1}{{ an x}} + C hfill \ end{gathered} ]$

448.

$[egin{gathered} int {frac{{{x^7}}}{{{{left( {1 - {x^2}} ight)}^5}}}{kern 1pt} { ext{d}}x} = - int {frac{{{x^7}}}{{{{left( {{x^2} - 1} ight)}^5}}}{kern 1pt} { ext{d}}x} hfill \ u = frac{1}{{{x^2} - 1}},frac{{{ ext{d}}u}}{{{ ext{d}}x}} = - frac{{2x}}{{{{left( {{x^2} - 1} ight)}^2}}}, hfill \ {x^6} = {left( {frac{1}{u} + 1} ight)^3},frac{1}{{{{left( {{x^2} - 1} ight)}^3}}} = {u^3}, hfill \ = frac{1}{2}int {{{left( {u + 1} ight)}^3}{kern 1pt} { ext{d}}u} = frac{{{{left( {u + 1} ight)}^4}}}{8} hfill \ = frac{{{{left( {frac{1}{{{x^2} - 1}} + 1} ight)}^4}}}{8} + C hfill \ end{gathered} ]$

449.

$[egin{gathered} int {frac{{arctan x}}{{{{left( {1 + {x^2}} ight)}^2}}}} {kern 1pt} { ext{d}}x hfill \ = int {frac{{arctan x}}{{{x^4} + 2{x^2} + 1}}{kern 1pt} { ext{d}}x} hfill \ u = arctan x,frac{{{ ext{d}}u}}{{{ ext{d}}x}} = frac{1}{{{x^2} + 1}},{x^2} = { an ^2}u, hfill \ {x^4} = { an ^4}u,{ an ^2}x = {sec ^2}x - 1, hfill \ {sec ^2}x = { an ^2}x + 1,frac{{{{ an }^2}x + 1}}{{{{ an }^4}x + 2{{ an }^2}x + 1}} = {cos ^2}x, hfill \ = int {u{{cos }^2}u{kern 1pt} { ext{d}}u} = int {uleft( {frac{{cos 2u}}{2} + frac{1}{2}} ight){kern 1pt} { ext{d}}u} hfill \ = frac{1}{2}int {uleft( {cos 2u + 1} ight){kern 1pt} { ext{d}}u} hfill \ = frac{1}{2}left( {int {ucos 2u{kern 1pt} { ext{d}}u} + int {u{kern 1pt} { ext{d}}u} } ight) hfill \ f = { ext{u}},g = cos 2u, hfill \ f = 1,g = frac{{sin 2u}}{2}, hfill \ end{gathered} ]$

$[egin{gathered} = frac{1}{2}left( {frac{{usin 2u}}{2} - int {frac{{sin 2u}}{2}{kern 1pt} { ext{d}}u} + frac{{{u^2}}}{2}} ight) hfill \ = frac{1}{2}left( {frac{{usin 2u}}{2} + frac{{cos 2u}}{4} + frac{{{u^2}}}{2}} ight) hfill \ = frac{{usin 2u}}{4} + frac{{cos 2u}}{8} + frac{{{u^2}}}{4} hfill \ sin 2u = sin left( {2arctan x} ight) = frac{{2x}}{{{x^2} + 1}}, hfill \ cos 2u = cos left( {arctan x} ight) = frac{{1 - {x^2}}}{{{x^2} + 1}}, hfill \ = frac{{{{arctan }^2}x}}{4} + frac{{xarctan x}}{{2left( {{x^2} + 1} ight)}} + frac{{1 - {x^2}}}{{8left( {{x^2} + 1} ight)}} + C hfill \ = frac{{left( {2{x^2} + 2} ight){{arctan }^2}x + 4xarctan x - {x^2} + 1}}{{8{x^2} + 8}} + C hfill \ end{gathered} ]$

450.

$[egin{gathered} int {frac{{{x^3}}}{{1 - {x^2}}}{kern 1pt} { ext{d}}x} = - int {frac{{{x^3}}}{{{x^2} - 1}}{kern 1pt} { ext{d}}x} hfill \ u = {x^2} - 1,frac{{{ ext{d}}u}}{{{ ext{d}}x}} = 2x, hfill \ = - frac{1}{2}int {frac{{u + 1}}{u}{kern 1pt} { ext{d}}u} hfill \ = - left( {frac{{ln u}}{2} + frac{u}{2}} ight) hfill \ = - frac{{ln left| {{x^2} - 1} ight|}}{2} - frac{{{x^2} - 1}}{2} + C hfill \ = - frac{{ln left| {{x^2} - 1} ight| + {x^2}}}{2} + C hfill \ end{gathered} ]$

451.

$[egin{gathered} int {frac{{ - {x^2} - 2}}{{{{left( {{x^2} + x + 1} ight)}^2}}}{kern 1pt} { ext{d}}x} = - int {frac{{{x^2} + 2}}{{{{left( {{x^2} + x + 1} ight)}^2}}}{kern 1pt} { ext{d}}x} hfill \ = - left[ {int {frac{1}{{{x^2} + x + 1}}{kern 1pt} { ext{d}}x} - int {frac{{x - 1}}{{{{left( {{x^2} + x + 1} ight)}^2}}}{kern 1pt} { ext{d}}x} } ight] hfill \ x - 1 = frac{1}{2}left( {2x + 1} ight) - frac{3}{2}, hfill \ = - left[ {int {frac{1}{{{{left( {x + frac{1}{2}} ight)}^2} + frac{3}{4}}}{kern 1pt} { ext{d}}x} - frac{1}{2}int {frac{{2x + 1}}{{{{left( {{x^2} + x + 1} ight)}^2}}}{kern 1pt} { ext{d}}x} + frac{3}{2}int {frac{1}{{{{left( {{x^2} + x + 1} ight)}^2}}}{kern 1pt} { ext{d}}x} } ight] hfill \ end{gathered} ]$

$[egin{gathered} = - left[ {frac{{2arctan frac{{2x + 1}}{{sqrt 3 }}}}{{sqrt 3 }} + frac{1}{2}frac{1}{{{x^2} + x + 1}} + frac{3}{2} imes 16int {frac{1}{{{{left( {{{left( {2x + 1} ight)}^2} + 3} ight)}^2}}}{kern 1pt} { ext{d}}x} } ight] hfill \ ...... hfill \ = - frac{{4arctan frac{{2x + 1}}{{sqrt 3 }}}}{{sqrt 3 }} - frac{{2left( {2x + 1} ight)}}{{{{left( {2x + 1} ight)}^2} + 3}} - frac{1}{{2left( {{x^2} + x + 1} ight)}} + C hfill \ = - frac{{4arctan frac{{2x + 1}}{{sqrt 3 }}}}{{sqrt 3 }} - frac{{x + 1}}{{{x^2} + x + 1}} + C hfill \ end{gathered} ]$

452.

$[egin{gathered} int_a^t {alpha (s)} left( {int_a^s {alpha (xi )x(xi )} dxi } ight){e^{int_s^t {alpha (u)} du}}ds hfill \ = - int_a^t {left( {int_a^s {alpha (xi )x(xi )} dxi } ight)} left( {frac{d}{{ds}}{e^{int_s^t {alpha (u)} du}}} ight)ds hfill \ = - left[ {left( {int_a^s {alpha (xi )x(xi )} dxi } ight){e^{int_s^t {alpha (u)} du}}} ight]_a^t + int_a^t {alpha (s)x(s)} {e^{int_s^t {alpha (u)} du}}ds hfill \ = - left( {int_a^t {alpha (xi )x(xi )} dxi } ight){e^{int_t^t {alpha (u)} du}} + left( {int_a^a {alpha (xi )x(xi )} dxi } ight){e^{int_a^t {alpha (u)} du}} + int_a^t {alpha (s)x(s)} {e^{int_s^t {alpha (u)} du}}ds hfill \ = - int_a^t {alpha (xi )x(xi )} dxi + int_a^t {alpha (s)x(s)} {e^{int_s^t {alpha (u)} du}}ds hfill \ end{gathered} ]$

453.

$[egin{gathered} int {{x^3}} sqrt {1 + {x^2}} {kern 1pt} { ext{d}}x hfill \ u = {x^2} + 1, hfill \ = frac{1}{2}int {left( {{u^{frac{3}{2}}} - sqrt u } ight){ ext{d}}u} hfill \ = frac{1}{2}left( {frac{{2{u^{frac{5}{2}}}}}{5} - frac{{2{u^{frac{3}{2}}}}}{3}} ight) hfill \ = frac{{{{left( {{x^2} + 1} ight)}^{frac{5}{2}}}}}{5} - frac{{{{left( {{x^2} + 1} ight)}^{frac{3}{2}}}}}{3} + C hfill \ end{gathered} ]$

454.

$[egin{gathered} int {frac{1}{{x + sqrt x }}{kern 1pt} { ext{d}}x} hfill \ = int {frac{{sqrt x - x}}{{left( {x + sqrt x } ight)left( {sqrt x - x} ight)}}{kern 1pt} { ext{d}}x} hfill \ = int {frac{{x - sqrt x }}{{{x^2} - x}}{kern 1pt} { ext{d}}x} hfill \ u = sqrt x ,frac{{{ ext{d}}u}}{{{ ext{d}}x}} = frac{1}{{2sqrt x }},x = {u^2},{x^2} = {u^4}, hfill \ = 2int {frac{1}{{u + 1}}{kern 1pt} { ext{d}}u} = 2int {frac{1}{{u + 1}}{kern 1pt} { ext{d}}left( {u + 1} ight)} hfill \ = 2ln left( {u + 1} ight) hfill \ = 2ln left( {sqrt x + 1} ight) + C hfill \ end{gathered} ]$

455.

$[egin{gathered} int {sqrt {{{sin }^2}x - {{sin }^4}x} {kern 1pt} { ext{d}}x} hfill \ = int {cos xsin x{kern 1pt} { ext{d}}x} hfill \ u = sin x,frac{{{ ext{d}}u}}{{{ ext{d}}x}} = cos x, hfill \ = int {u{kern 1pt} { ext{d}}u} = frac{{{u^2}}}{2} hfill \ = frac{{{{sin }^2}x}}{2} + C hfill \ int_0^{frac{pi }{2}} {sqrt {{{sin }^2}x - {{sin }^4}x} {kern 1pt} { ext{d}}x = frac{1}{2}} hfill \ end{gathered} ]$

456.

$[egin{gathered} int {{{cos }^2}x{{sin }^2}x{kern 1pt} { ext{d}}x} hfill \ = int {left( {{{sin }^2}x - {{sin }^4}x} ight){ ext{d}}x} hfill \ = int {{{sin }^2}x{kern 1pt} { ext{d}}x} - int {{{sin }^4}x{kern 1pt} { ext{d}}x} hfill \ = int {frac{{1 - cos 2x}}{2}{kern 1pt} { ext{d}}x} - int {frac{{cos 4x - 4cos 2x + 3}}{8}{kern 1pt} { ext{d}}x} hfill \ = frac{1}{2}int {1{kern 1pt} { ext{d}}x} - frac{1}{2}int {cos 2x{kern 1pt} { ext{d}}x} - left[ {frac{{sin 4x}}{{32}} - frac{{sin 2x}}{4} + frac{{3x}}{8}} ight] hfill \ = - frac{{sin 4x - 4x}}{{32}} + C hfill \ int_0^{frac{pi }{2}} {{{cos }^2}x{{sin }^2}x{kern 1pt} { ext{d}}x} = frac{pi }{{16}} hfill \ end{gathered} ]$

457.

$[egin{gathered} int {frac{{{{ ext{e}}^{frac{1}{x}}}}}{{{x^2}}}{kern 1pt} { ext{d}}x} hfill \ u = frac{1}{x},frac{{{ ext{d}}u}}{{{ ext{d}}x}} = - frac{1}{{{x^2}}}, hfill \ = - int {{{ ext{e}}^u}{kern 1pt} { ext{d}}u} hfill \ = - {{ ext{e}}^u} hfill \ = - {{ ext{e}}^{frac{1}{x}}} + C hfill \ end{gathered} ]$

458.

$[egin{gathered} int {frac{{x - 1}}{{{x^2} + 1}}{kern 1pt} { ext{d}}x} hfill \ = int {left( {frac{x}{{{x^2} + 1}} - frac{1}{{{x^2} + 1}}} ight){ ext{d}}x} hfill \ = int {frac{x}{{{x^2} + 1}}{kern 1pt} { ext{d}}x} - int {frac{1}{{{x^2} + 1}}{kern 1pt} { ext{d}}x} hfill \ = frac{1}{2}int {frac{1}{{{x^2} + 1}}{kern 1pt} { ext{d}}left( {{x^2} + 1} ight)} - arctan x hfill \ = frac{{ln left( {{x^2} + 1} ight)}}{2} - arctan x + C hfill \ end{gathered} ]$

459.

$[egin{gathered} mathop {lim }limits_{t o + infty } left( { - frac{1}{s}{e^{ - st}}} ight) = - frac{1}{s}mathop {lim }limits_{t o + infty } {e^{ - st}} hfill \ = - frac{1}{s}mathop {lim }limits_{t o + infty } frac{1}{{{e^{st}}}} hfill \ s = x + iy, hfill \ mathop {lim }limits_{t o + infty } left( { - frac{1}{s}{e^{ - st}}} ight) = - frac{1}{s}mathop {lim }limits_{t o + infty } frac{1}{{{e^{left( {x + iy} ight)t}}}} = - frac{1}{s}mathop {lim }limits_{t o + infty } frac{1}{{{e^{xt + iyt}}}} hfill \ = - frac{1}{s}mathop {lim }limits_{t o + infty } frac{1}{{{e^{xt}}{e^{iyt}}}} hfill \ {e^{iyt}} = cos yt + isin yt, hfill \ mathop {lim }limits_{t o + infty } left( { - frac{1}{s}{e^{ - st}}} ight) = - frac{1}{s}mathop {lim }limits_{t o + infty } frac{1}{{{e^{xt}}left( {cos yt + isin yt} ight)}} hfill \ = - frac{1}{s}mathop {lim }limits_{t o + infty } frac{{cos yt - isin yt}}{{{e^{xt}}left( {cos yt + isin yt} ight)left( {cos yt - isin yt} ight)}} hfill \ = - frac{1}{s}mathop {lim }limits_{t o + infty } frac{{cos yt - isin yt}}{{{e^{xt}}left( {{{cos }^2}yt + {{sin }^2}yt} ight)}} hfill \ = - frac{1}{s}mathop {lim }limits_{t o + infty } frac{{cos yt - isin yt}}{{{e^{xt}}}} = - frac{1}{s}mathop {lim }limits_{t o + infty } left( {frac{{cos yt}}{{{e^{xt}}}} - ifrac{{sin yt}}{{{e^{xt}}}}} ight) hfill \ = - frac{1}{s}left( {0 - i0} ight) = 0 hfill \ end{gathered} ]$

460.

$[egin{gathered} int_0^{ + infty } {{x^{frac{3}{2}}}} {e^{ - x}}dx hfill \ = - int_0^{ + infty } {{x^{frac{3}{2}}}} d{e^{ - x}} hfill \ = - left( {{x^{frac{3}{2}}}{e^{ - x}}|_0^{ + infty } - int_0^{ + infty } {{e^{ - x}}} d{x^{frac{3}{2}}}} ight) hfill \ = - {x^{frac{3}{2}}}{e^{ - x}}|_0^{ + infty } + frac{3}{2}int_0^{ + infty } {{x^{frac{1}{2}}}{e^{ - x}}} dx hfill \ = - {x^{frac{3}{2}}}{e^{ - x}}|_0^{ + infty } - frac{3}{2}int_0^{ + infty } {{x^{frac{1}{2}}}} d{e^{ - x}} hfill \ = - {x^{frac{3}{2}}}{e^{ - x}}|_0^{ + infty } - frac{3}{2}left( {{x^{frac{1}{2}}}{e^{ - x}}|_0^{ + infty } - int_0^{ + infty } {{e^{ - x}}} d{x^{frac{1}{2}}}} ight) hfill \ = - {x^{frac{3}{2}}}{e^{ - x}}|_0^{ + infty } - frac{3}{2}left( {{x^{frac{1}{2}}}{e^{ - x}}|_0^{ + infty } - frac{1}{2}int_0^{ + infty } {{x^{ - frac{1}{2}}}{e^{ - x}}} dx} ight) hfill \ = - {x^{frac{3}{2}}}{e^{ - x}}|_0^{ + infty } - frac{3}{2}{x^{frac{1}{2}}}{e^{ - x}}|_0^{ + infty } + frac{3}{4}int_0^{ + infty } {{x^{ - frac{1}{2}}}{e^{ - x}}} dx hfill \ {x^{frac{1}{2}}} = t,dx = 2tdt, hfill \ = - {x^{frac{3}{2}}}{e^{ - x}}|_0^{ + infty } - frac{3}{2}{x^{frac{1}{2}}}{e^{ - x}}|_0^{ + infty } + frac{3}{2}int_0^{ + infty } {{e^{ - {t^2}}}} dt hfill \ I = int_0^{ + infty } {{e^{ - {t^2}}}} dt, hfill \ {I^2} = int_0^{ + infty } {{e^{ - {t^2}}}} dtint_0^{ + infty } {{e^{ - {z^2}}}} dz hfill \ = int_0^{ + infty } {int_0^{ + infty } {{e^{ - ({t^2} + {z^2})}}} dzdt} hfill \ = int_0^{frac{pi }{2}} {d heta } int_0^{ + infty } {{e^{ - {r^2}}}rdr} = frac{pi }{4} hfill \ herefore I = frac{{sqrt pi }}{2}, hfill \ herefore int_0^{ + infty } {{x^{frac{3}{2}}}} {e^{ - x}}dx hfill \ = - {x^{frac{3}{2}}}{e^{ - x}}|_0^{ + infty } - frac{3}{2}{x^{frac{1}{2}}}{e^{ - x}}|_0^{ + infty } + frac{3}{2}int_0^{ + infty } {{e^{ - {t^2}}}} dt hfill \ = - {x^{frac{3}{2}}}{e^{ - x}}|_0^{ + infty } - frac{3}{2}{x^{frac{1}{2}}}{e^{ - x}}|_0^{ + infty } + frac{{3sqrt pi }}{4} hfill \ = frac{{3sqrt pi }}{4} hfill \ end{gathered} ]$

461.

$[ Rightarrow {k_1}(x - {x_1}) = {k_2}y, Rightarrow {k_1}(mathop xlimits^. - mathop {{x_1}}limits^. ) = {k_2}mathop ylimits^. Rightarrow mathop {{x_1}}limits^. = mathop xlimits^. - frac{{{k_2}}}{{{k_1}}}mathop ylimits^. ]$

代入3-5，得所求微分方程.

462.

$[egin{gathered} int {frac{1}{{sqrt {xleft( {1 + x} ight)} }}{kern 1pt} { ext{d}}x} = int {frac{1}{{sqrt x sqrt {x + 1} }}{kern 1pt} { ext{d}}x} hfill \ u = sqrt x ,{ ext{d}}x = 2sqrt x {kern 1pt} { ext{d}}u, hfill \ = 2int {frac{1}{{sqrt {{u^2} + 1} }}{kern 1pt} { ext{d}}u} hfill \ u = an v,v = arctan u,{ ext{d}}u = {sec ^2}v{kern 1pt} { ext{d}}v, hfill \ = 2int {frac{{{{sec }^2}v}}{{sqrt {{{ an }^2}v + 1} }}{kern 1pt} { ext{d}}v} = 2int {sec v{ ext{d}}v} hfill \ = 2ln left( { an v + sec v} ight) hfill \ = 2ln left( {sqrt {{u^2} + 1} + u} ight) hfill \ = 2ln left( {sqrt {x + 1} + sqrt x } ight) + C hfill \ = ln left| {2sqrt {{x^2} + x} + 2x + 1} ight| + C hfill \ end{gathered} ]$

463.

$[egin{gathered} int {frac{1}{{sqrt {4 - x} }}{kern 1pt} { ext{d}}x} hfill \ { ext{u}} = { ext{4}} - { ext{x}}, hfill \ = - int {frac{1}{{sqrt u }}{kern 1pt} { ext{d}}u} hfill \ = - 2sqrt u hfill \ = - 2sqrt {4 - x} + C hfill \ end{gathered} ]$

464.

$[egin{gathered} int {frac{{sqrt {x - 1} }}{{{x^2}}}{kern 1pt} { ext{d}}x} hfill \ u = sqrt {x - 1} ,{ ext{d}}x = 2sqrt {x - 1} {kern 1pt} { ext{d}}u, hfill \ = 2int {frac{{{u^2}}}{{{{left( {{u^2} + 1} ight)}^2}}}{kern 1pt} { ext{d}}u} hfill \ = 2int {frac{{{u^2} + 1 - 1}}{{{{left( {{u^2} + 1} ight)}^2}}}{kern 1pt} { ext{d}}u} hfill \ = 2left[ {int {frac{1}{{{u^2} + 1}}{kern 1pt} { ext{d}}u} - int {frac{1}{{{{left( {{u^2} + 1} ight)}^2}}}{kern 1pt} { ext{d}}u} } ight] hfill \ = 2left[ {arctan u - left( {frac{1}{2}int {frac{1}{{{u^2} + 1}}{kern 1pt} { ext{d}}u} + frac{u}{{2left( {{u^2} + 1} ight)}}} ight)} ight] hfill \ = arctan u - frac{u}{{{u^2} + 1}} hfill \ = arctan sqrt {x - 1} - frac{{sqrt {x - 1} }}{x} + C hfill \ end{gathered} ]$

465.

$[egin{gathered} int {{{left( {2x + 1} ight)}^3}{kern 1pt} { ext{d}}x} hfill \ { ext{u}} = { ext{2x}} + { ext{1}}, hfill \ = frac{1}{2}int {{u^3}{kern 1pt} { ext{d}}u} = frac{{{u^4}}}{8} hfill \ = frac{{{{left( {2x + 1} ight)}^4}}}{8} + C hfill \ end{gathered} ]$

466.

$[egin{gathered} left| {frac{{sin x}}{{sqrt x }} - 0} ight| = frac{{left| {sin x} ight|}}{{sqrt x }} leqslant frac{1}{{sqrt x }} < varepsilon Rightarrow sqrt x > frac{1}{varepsilon } Rightarrow x > frac{1}{{{varepsilon ^2}}} hfill \ forall varepsilon > 0,exists X = frac{1}{{{varepsilon ^2}}} > 0, hfill \ end{gathered} ]$

當 $[x > X]$ 時，就有 $[left| {frac{{sin x}}{{sqrt x }} - 0} ight| < varepsilon ]$ ，所以 $[mathop {lim }limits_{x o + infty } frac{{sin x}}{{sqrt x }}{ ext{ = }}0]$

467.

$[egin{gathered} mathop {lim }limits_{x o 0} frac{{ an x - sin x}}{x} = mathop {lim }limits_{x o 0} frac{{frac{{sin x}}{{cos x}} - frac{{sin xcos x}}{{cos x}}}}{x} hfill \ = mathop {lim }limits_{x o 0} frac{{sin x - sin xcos x}}{{xcos x}} = mathop {lim }limits_{x o 0} frac{{sin xleft( {1 - cos x} ight)}}{{xcos x}} hfill \ = mathop {lim }limits_{x o 0} frac{{sin x}}{x}frac{{1 - cos x}}{{cos x}} = mathop {lim }limits_{x o 0} frac{{1 - cos x}}{{cos x}} = 0 hfill \ # # hfill \ mathop {lim }limits_{x o 0} frac{{1 - cos x}}{{{x^2}cos x}} = mathop {lim }limits_{x o 0} frac{{2{{left( {sin frac{x}{2}} ight)}^2}}}{{4{{left( {frac{x}{2}} ight)}^2}cos x}} = mathop {lim }limits_{x o 0} frac{1}{2}{left( {frac{{sin frac{x}{2}}}{{frac{x}{2}}}} ight)^2}frac{1}{{cos x}} = frac{1}{2} e 0 hfill \ herefore mathop {lim }limits_{x o 0} frac{{ an x - sin x}}{{{x^3}}} = frac{1}{2} e 0 hfill \ end{gathered} ]$

$[{ an x - sin x}]$ 是關於 $x$ 的3階無窮小.

468.

$[egin{gathered} mathop {lim }limits_{x o infty } frac{{{{left( {4{x^2} - 3} ight)}^3}{{left( {3x - 2} ight)}^4}}}{{{{left( {6{x^2} + 7} ight)}^5}}} = mathop {lim }limits_{x o infty } frac{{{x^{10}}}}{{{x^{10}}}}frac{{{{left( {4{x^2} - 3} ight)}^3}{{left( {3x - 2} ight)}^4}}}{{{{left( {6{x^2} + 7} ight)}^5}}} hfill \ = mathop {lim }limits_{x o infty } frac{{frac{1}{{{x^{10}}}}}}{{frac{1}{{{x^{10}}}}}}frac{{{{left( {4{x^2} - 3} ight)}^3}{{left( {3x - 2} ight)}^4}}}{{{{left( {6{x^2} + 7} ight)}^5}}} = mathop {lim }limits_{x o infty } frac{{{{left( {frac{1}{{{x^2}}}} ight)}^3}{{left( {4{x^2} - 3} ight)}^3}frac{1}{{{x^4}}}{{left( {3x - 2} ight)}^4}}}{{{{left( {frac{1}{{{x^2}}}} ight)}^5}{{left( {6{x^2} + 7} ight)}^5}}} hfill \ = mathop {lim }limits_{x o infty } frac{{{{left( {frac{{4{x^2} - 3}}{{{x^2}}}} ight)}^3}{{left( {frac{{3x - 2}}{x}} ight)}^4}}}{{{{left( {frac{{6{x^2} + 7}}{{{x^2}}}} ight)}^5}}} = mathop {lim }limits_{x o infty } frac{{{{left( {4 - frac{3}{{{x^2}}}} ight)}^3}{{left( {3 - frac{2}{x}} ight)}^4}}}{{{{left( {6 + frac{7}{{{x^2}}}} ight)}^5}}} hfill \ = frac{{{4^3}{3^4}}}{{{6^5}}} = frac{{4 imes 4 imes 4 imes 3 imes 3 imes 3 imes 3}}{{6 imes 6 imes 6 imes 6 imes 6}} = frac{2}{3} hfill \ # # # hfill \ mathop {lim }limits_{x o infty } frac{{x + sin x}}{{x - cos x}} = mathop {lim }limits_{x o infty } frac{{1 + frac{1}{x}sin x}}{{1 - frac{1}{x}cos x}} = mathop {lim }limits_{x o infty } frac{{1 + 0}}{{1 - 0}} = 1 hfill \ end{gathered} ]$

469.

$[egin{gathered} mathop {lim }limits_{x o 0} frac{{sin 2x - 2sin x}}{x} = mathop {lim }limits_{x o 0} frac{{2sin xcos x - 2sin x}}{x} hfill \ = mathop {lim }limits_{x o 0} frac{{2sin xleft( {cos x - 1} ight)}}{x} = 2mathop {lim }limits_{x o 0} frac{{sin x}}{x}mathop {lim }limits_{x o 0} left( {cos x - 1} ight) hfill \ = 2mathop {lim }limits_{x o 0} left( {cos x - 1} ight) = 0 hfill \ ecause cos x = 1 - 2{sin ^2}frac{x}{2}, hfill \ herefore 2mathop {lim }limits_{x o 0} frac{{left( {cos x - 1} ight)}}{{{x^2}}} = 2mathop {lim }limits_{x o 0} frac{{2{{sin }^2}frac{x}{2}}}{{{x^2}}} = 4mathop {lim }limits_{x o 0} frac{{{{left( {sin frac{x}{2}} ight)}^2}}}{{4{{left( {frac{x}{2}} ight)}^2}}} hfill \ = mathop {lim }limits_{x o 0} {left( {frac{{sin frac{x}{2}}}{{frac{x}{2}}}} ight)^2} = 1 hfill \ herefore mathop {lim }limits_{x o 0} frac{{sin 2x - 2sin x}}{{{x^3}}} = 1 e 0 hfill \ end{gathered} ]$

$[{sin 2x - 2sin x}]$ 是關於 $x$ 的3階無窮小.

470.

$[egin{gathered} mathop {lim }limits_{x o infty } left( {frac{{{x^2} + 2}}{{x + 1}} - ax - b} ight) = 0 hfill \ mathop {lim }limits_{x o infty } left( {frac{{{x^2} + 2}}{{x + 1}} - frac{{a{x^2} + ax}}{{x + 1}} - frac{{bx + b}}{{x + 1}}} ight) = 0 hfill \ mathop {lim }limits_{x o infty } frac{{x + frac{2}{x} - ax - a - b - frac{b}{x}}}{{1 + frac{1}{x}}} = 0 hfill \ mathop {lim }limits_{x o infty } frac{{x + 0 - ax - a - b - 0}}{{1 + 0}} = 0 hfill \ mathop {lim }limits_{x o infty } left[ {xleft( {1 - a} ight) - a - b} ight] = 0 hfill \ herefore a = 1,b = - 1 hfill \ end{gathered} ]$

471.

$eqalign{ & mathop {lim }limits_{x o infty } frac{{x - sin x}}{{x + sin x}} = mathop {lim }limits_{x o infty } frac{{1 - frac{1}{x}sin x}}{{1 + frac{1}{x}sin x}} cr & = frac{{1 - mathop {lim }limits_{x o infty } frac{1}{x}sin x}}{{1 + mathop {lim }limits_{x o infty } frac{1}{x}sin x}} = frac{{1 - 0}}{{1 + 0}} = 1 cr & (mathop {lim }limits_{x o infty } frac{1}{x}sin x = 0,wuqiongxiaoyuyoujiehanshudechengjishiwuqiongxiao) cr & mathop {lim }limits_{x o 0} {left( {1 - frac{x}{2}} ight)^{frac{1}{x}}} = mathop {lim }limits_{x o 0} {left( {1 - frac{x}{2}} ight)^{ - frac{2}{x}left( { - frac{1}{2}} ight)}} = mathop {lim }limits_{x o 0} {left( {1 - frac{x}{2}} ight)^{ - frac{2}{x}}}{left( {1 - frac{x}{2}} ight)^{ - frac{1}{2}}} cr & = mathop {lim }limits_{x o 0} {left( {1 - frac{x}{2}} ight)^{ - frac{2}{x}}}{left( {1 - frac{x}{2}} ight)^{ - frac{1}{2}}} = { ext{e}} cr & mathop {lim }limits_{x o infty } {left( {frac{{x - 1}}{{x + 1}}} ight)^{2x}} cr & u = x + 1,x = u - 1,x - 1 = u - 2, cr & mathop {lim }limits_{x o infty } {left( {frac{{x - 1}}{{x + 1}}} ight)^{2x}} = mathop {lim }limits_{x o infty } {left( {frac{{u - 2}}{u}} ight)^{2u - 2}} cr & = mathop {lim }limits_{x o infty } {left[ {{{left( {1 - frac{2}{u}} ight)}^u}} ight]^2}{left( {1 - frac{2}{u}} ight)^{ - 2}} cr & = mathop {lim }limits_{x o infty } {left[ {{{left( {1 - frac{2}{u}} ight)}^{ - frac{u}{2}left( { - 2} ight)}}} ight]^2}{left( {1 - frac{2}{u}} ight)^{ - 2}} cr & = mathop {lim }limits_{x o infty } {left[ {{{left( {1 - frac{2}{u}} ight)}^{ - frac{u}{2}}}{{left( {1 - frac{2}{u}} ight)}^{ - 2}}} ight]^2}{left( {1 - frac{2}{u}} ight)^{ - 2}} cr & = {left[ {{ ext{e}} imes 1} ight]^2} imes 1{ ext{ = }}{{ ext{e}}^2} cr}$

472.

$eqalign{ & mathop {lim }limits_{x o + infty } left[ {{{left( {{x^3} + {x^2} + x + 1} ight)}^{frac{1}{3}}} - x} ight] = mathop {lim }limits_{x o + infty } left[ { oot 3 of {{x^3} + {x^2} + x + 1} - x} ight] cr & = mathop {lim }limits_{x o + infty } xleft[ {frac{{left( { oot 3 of {{x^3} + {x^2} + x + 1} - x} ight)}}{x}} ight] = mathop {lim }limits_{x o + infty } frac{{ oot 3 of {1 + frac{1}{x} + frac{1}{{{x^2}}} + frac{1}{{{x^3}}}} - 1}}{{frac{1}{x}}} cr & = mathop {lim }limits_{x o + infty } frac{{frac{{ - frac{1}{{{x^2}}} - frac{1}{{{x^3}}} - frac{1}{{{x^4}}}}}{{3{{left( {1 + frac{1}{x} + frac{1}{{{x^2}}} + frac{1}{{{x^3}}}} ight)}^{frac{2}{3}}}}}}}{{ - frac{1}{{{x^2}}}}} = mathop {lim }limits_{x o + infty } frac{{1 + frac{1}{x} + frac{1}{{{x^2}}}}}{{3{{left( {1 + frac{1}{x} + frac{1}{{{x^2}}} + frac{1}{{{x^3}}}} ight)}^{frac{2}{3}}}}} cr & = frac{1}{3} cr}$

473.

$eqalign{ & y = xsqrt {1 - {x^2}} + arcsin x cr & y = sqrt {1 - {x^2}} + frac{{ - 2x}}{{2sqrt {1 - {x^2}} }}x + frac{1}{{sqrt {1 - {x^2}} }} cr & = sqrt {1 - {x^2}} - frac{{{x^2}}}{{sqrt {1 - {x^2}} }} + frac{1}{{sqrt {1 - {x^2}} }} cr & = frac{{2left( {1 - {x^2}} ight)}}{{sqrt {1 - {x^2}} }} = frac{{2left( {1 - {x^2}} ight)sqrt {1 - {x^2}} }}{{1 - {x^2}}} cr & = 2sqrt {1 - {x^2}} cr & cr & y = {{ ext{e}}^{ - {x^2}cos frac{1}{x}}} cr & ln y = - {x^2}cos frac{1}{x}ln { ext{e}} = - {x^2}cos frac{1}{x} cr & frac{{y}}{y} = - left[ {2xcos frac{1}{x} + left( { - sin frac{1}{x}} ight)left( { - frac{1}{{{x^2}}}} ight){x^2}} ight] cr & = - 2xcos frac{1}{x} - sin frac{1}{x} cr & y = left( { - 2xcos frac{1}{x} - sin frac{1}{x}} ight){{ ext{e}}^{ - {x^2}cos frac{1}{x}}} cr}$

474.

$[egin{gathered} left{ egin{gathered} x - {e^x}sin t + 1 = 0 hfill \ y = {t^3} + 2t hfill \ end{gathered} ight. Rightarrow left{ egin{gathered} frac{{dx}}{{dt}} - {e^x}frac{{dx}}{{dt}}sin t - {e^x}cos t = 0 hfill \ frac{{dy}}{{dt}} = 3{t^2} + 2 hfill \ end{gathered} ight. hfill \ Rightarrow left{ egin{gathered} frac{{dx}}{{dt}}left( {1 - {e^x}sin t} ight) = {e^x}cos t Rightarrow frac{{dx}}{{dt}} = frac{{{e^x}cos t}}{{1 - {e^x}sin t}} hfill \ frac{{dy}}{{dt}} = 3{t^2} + 2 hfill \ end{gathered} ight. hfill \ frac{{dy}}{{dx}} = frac{{dy}}{{dt}}frac{{dt}}{{dx}} = left( {3{t^2} + 2} ight)frac{{1 - {e^x}sin t}}{{{e^x}cos t}} hfill \ frac{{dy}}{{dx}}{|_{t = 0}} = 2left( {frac{1}{{{e^x}}}} ight) = frac{2}{{{e^x}}} hfill \ end{gathered} ]$

475.

$[egin{gathered} int {frac{1}{{{{left( {{x^2} + x + 1} ight)}^2}}}{kern 1pt} { ext{d}}x} hfill \ = int {frac{1}{{{{left[ {{{left( {x + frac{1}{2}} ight)}^2} + frac{3}{4}} ight]}^2}}}{kern 1pt} { ext{d}}x} hfill \ = 16int {frac{1}{{{{left[ {{{left( {2x + 1} ight)}^2} + 3} ight]}^2}}}{kern 1pt} { ext{d}}x} hfill \ { ext{u}} = { ext{2x}} + { ext{1}}, hfill \ = 16 imes frac{1}{2}int {frac{1}{{{{left( {{u^2} + 3} ight)}^2}}}{kern 1pt} { ext{d}}u} hfill \ = 8left[ {frac{1}{6}int {frac{1}{{{u^2} + 3}}{kern 1pt} { ext{d}}u} + frac{u}{{6left( {{u^2} + 3} ight)}}} ight] hfill \ v = frac{u}{{sqrt 3 }},{ ext{d}}u = sqrt 3 {kern 1pt} { ext{d}}v, hfill \ = 8left[ {frac{1}{6} imes frac{1}{{sqrt 3 }}int {frac{1}{{{v^2} + 1}}{kern 1pt} { ext{d}}v} + frac{u}{{6left( {{u^2} + 3} ight)}}} ight] hfill \ = 8left[ {frac{1}{6} imes frac{{arctan frac{u}{{sqrt 3 }}}}{{sqrt 3 }} + frac{u}{{6left( {{u^2} + 3} ight)}}} ight] hfill \ = frac{{4sqrt 3 arctan frac{{2x + 1}}{{sqrt 3 }}}}{3} + frac{{2x + 1}}{{3left( {{x^2} + x + 1} ight)}} + C hfill \ end{gathered} ]$

476.

$[egin{gathered} y = sqrt {xsin xsqrt {x + {{ ext{e}}^{2x}}} } hfill \ ln y = frac{1}{2}ln left[ {xsin x{{left( {x + {{ ext{e}}^{2x}}} ight)}^{frac{1}{2}}}} ight] hfill \ frac{{y}}{y} = frac{1}{2}frac{{left[ {xsin x{{left( {x + {{ ext{e}}^{2x}}} ight)}^{frac{1}{2}}}} ight]}}{{xsin x{{left( {x + {{ ext{e}}^{2x}}} ight)}^{frac{1}{2}}}}} hfill \ = frac{1}{2}frac{{left[ {sin x{{left( {{x^3} + {x^2}{{ ext{e}}^{2x}}} ight)}^{frac{1}{2}}}} ight]}}{{xsin x{{left( {x + {{ ext{e}}^{2x}}} ight)}^{frac{1}{2}}}}} hfill \ = frac{1}{2}frac{{cos x{{left( {{x^3} + {x^2}{{ ext{e}}^{2x}}} ight)}^{frac{1}{2}}} + frac{{left[ {3{x^2} + left( {2x{{ ext{e}}^{2x}} + 2{x^2}{{ ext{e}}^{2x}}} ight)} ight]sin x}}{{2{{left( {{x^3} + {x^2}{{ ext{e}}^{2x}}} ight)}^{frac{1}{2}}}}}}}{{xsin x{{left( {x + {{ ext{e}}^{2x}}} ight)}^{frac{1}{2}}}}} hfill \ = frac{{2cos xleft( {{x^3} + {x^2}{{ ext{e}}^{2x}}} ight) + left( {3{x^2} + 2x{{ ext{e}}^{2x}} + 2{x^2}{{ ext{e}}^{2x}}} ight)sin x}}{{4sin xleft( {{x^3} + {x^2}{{ ext{e}}^{2x}}} ight)}} hfill \ y = frac{{2cos xleft( {{x^3} + {x^2}{{ ext{e}}^{2x}}} ight) + left( {3{x^2} + 2x{{ ext{e}}^{2x}} + 2{x^2}{{ ext{e}}^{2x}}} ight)sin x}}{{4sin xleft( {{x^3} + {x^2}{{ ext{e}}^{2x}}} ight)}}sqrt {xsin xsqrt {x + {{ ext{e}}^{2x}}} } hfill \ end{gathered} ]$

477.

$[egin{gathered} left{ egin{gathered} x = 2{{ ext{e}}^t} hfill \ y = {{ ext{e}}^{ - t}} hfill \ end{gathered} ight. hfill \ frac{{{ ext{d}}x}}{{{ ext{d}}t}} = 2{{ ext{e}}^t} hfill \ frac{{{ ext{d}}y}}{{{ ext{d}}x}} = frac{{{ ext{d}}y}}{{{ ext{d}}t}} cdot frac{{{ ext{d}}t}}{{{ ext{d}}x}} = frac{{{ ext{d}}y}}{{{ ext{d}}t}} cdot frac{1}{{frac{{{ ext{d}}x}}{{{ ext{d}}t}}}}{ ext{ = }}{{ ext{e}}^{ - t}} imes left( { - 1} ight) cdot frac{1}{{2{{ ext{e}}^t}}} hfill \ = - frac{1}{2}{{ ext{e}}^{ - 2t}} hfill \ frac{{{ ext{d}}y}}{{{ ext{d}}x}}{|_{t = 0}} = - frac{1}{2} hfill \ end{gathered} ]$

478.

$eqalign{ & y = {cos ^2}(arctan {x^3}) cr & u = arctan {x^3},v = {x^3}, cr & y = {cos ^2}u cr & y = 2cos u( - sin u)u cr & = - 2sin ucos ufrac{1}{{1 + {x^6}}} imes 3{x^2} cr & = frac{{ - 6{x^2}sin (arctan {x^3})cos (arctan {x^3})}}{{1 + {x^6}}} cr}$