積分習題及詳解16

03-29

35.

$eqalign{ & mathop {lim }limits_{x o + infty } frac{{{{left( {1 + frac{1}{x}} ight)}^{{x^2}}} + {{ ext{e}}^x}}}{x} cr & = mathop {lim }limits_{x o + infty } left[ {{{left( {1 + frac{1}{x}} ight)}^{{x^2}}} + {{ ext{e}}^x}} ight] cr & y = {left( {1 + frac{1}{x}} ight)^{{x^2}}}, cr & ln y = {x^2}ln left( {1 + frac{1}{x}} ight) cr & frac{{y}}{y} = 2xln left( {1 + frac{1}{x}} ight) + frac{{ - frac{{{x^2}}}{{{x^2}}}}}{{1 + frac{1}{x}}} cr & = 2xln left( {1 + frac{1}{x}} ight) - frac{1}{{1 + frac{1}{x}}} cr & y = left[ {2xln left( {1 + frac{1}{x}} ight) - frac{1}{{1 + frac{1}{x}}}} ight]{left( {1 + frac{1}{x}} ight)^{{x^2}}} cr & Rightarrow cr & = mathop {lim }limits_{x o + infty } left[ {2xln left( {1 + frac{1}{x}} ight) - frac{1}{{1 + frac{1}{x}}}} ight]{left( {1 + frac{1}{x}} ight)^{{x^2}}} + {{ ext{e}}^x} cr & mathop {lim }limits_{x o + infty } 2xln left( {1 + frac{1}{x}} ight) cr & = 2mathop {lim }limits_{x o + infty } frac{{ln left( {1 + frac{1}{x}} ight)}}{{frac{1}{x}}} cr & = 2mathop {lim }limits_{x o + infty } frac{{frac{{ - frac{1}{{{x^2}}}}}{{1 + frac{1}{x}}}}}{{ - frac{1}{{{x^2}}}}} cr & = 2mathop {lim }limits_{x o + infty } frac{1}{{1 + frac{1}{x}}} cr & = 2 cr & mathop {lim }limits_{x o + infty } {left( {1 + frac{1}{x}} ight)^{{x^2}}} cr & y = {left( {1 + frac{1}{x}} ight)^{{x^2}}} cr & ln y = {x^2}ln left( {1 + frac{1}{x}} ight) cr & y = {{ ext{e}}^{{x^2}ln left( {1 + frac{1}{x}} ight)}} cr & = mathop {lim }limits_{x o + infty } {{ ext{e}}^{{x^2}ln left( {1 + frac{1}{x}} ight)}} cr & = mathop {lim }limits_{x o + infty } {{ ext{e}}^{frac{{ln left( {1 + frac{1}{x}} ight)}}{{frac{1}{{{x^2}}}}}}} cr & = mathop {lim }limits_{x o + infty } {{ ext{e}}^{frac{{frac{{ - frac{1}{{{x^2}}}}}{{1 + frac{1}{x}}}}}{{ - 2frac{1}{{{x^3}}}}}}} cr & = mathop {lim }limits_{x o + infty } {{ ext{e}}^{frac{x}{{2left( {1 + frac{1}{x}} ight)}}}} cr & = + infty cr & herefore = + infty cr}$

34.

$eqalign{ & int {frac{x}{{sqrt {1 - {x^2}} }}{kern 1pt} { ext{d}}x} cr & u = 1 - {x^2},{ ext{d}}x = - frac{1}{{2x}}{kern 1pt} { ext{d}}u, cr & = - frac{1}{2}int {frac{1}{{sqrt u }}{kern 1pt} { ext{d}}u} cr & = - frac{1}{2} imes 2sqrt u cr & = - sqrt u cr & = - sqrt {1 - {x^2}} + C cr & intlimits_0^1 {frac{x}{{sqrt {1 - {x^2}} }}{kern 1pt} { ext{d}}x} cr & = 1 cr & # cr & int {x{{ ext{e}}^{ - x}}{kern 1pt} { ext{d}}x} cr & u = { ext{x}},v = {{ ext{e}}^{ - x}}, cr & u = 1,v = - {{ ext{e}}^{ - x}}, cr & = - int { - {{ ext{e}}^{ - x}}{kern 1pt} { ext{d}}x} - x{{ ext{e}}^{ - x}} cr & = - int { - {{ ext{e}}^{ - x}}{kern 1pt} { ext{d}}x} - x{{ ext{e}}^{ - x}} cr & = int { - {{ ext{e}}^{ - x}}{kern 1pt} { ext{d}}left( { - x} ight)} - x{{ ext{e}}^{ - x}} cr & = - {{ ext{e}}^{ - x}} - x{{ ext{e}}^{ - x}} + C cr & = - left( {x + 1} ight){{ ext{e}}^{ - x}} + C cr & intlimits_0^{ + infty } {x{{ ext{e}}^{ - x}}{kern 1pt} { ext{d}}x} cr & = - mathop {lim }limits_{x o + infty } frac{{x + 1}}{{{{ ext{e}}^x}}} + 1 cr & = - mathop {lim }limits_{x o + infty } frac{1}{{{{ ext{e}}^x}}} + 1 cr & = 1 cr}$