積分練習草稿7

02-13

211.

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

212.

$[egin{gathered} int {frac{{4{t^2}}}{{1 + {t^2}}}} {kern 1pt} { ext{d}}t = 4int {} frac{{{t^2}}}{{{t^2} + 1}}{kern 1pt} { ext{d}}t hfill \ {t^2}{ ext{ = }}{t^2} + 1 - 1, hfill \ = 4(int {1{kern 1pt} { ext{d}}t} - int {frac{1}{{{t^2} + 1}}{kern 1pt} { ext{d}}t} ) hfill \ = 4t - 4arctan t + C hfill \ end{gathered} ]$

213.

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

214.

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

215.

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

216.

$[egin{gathered} int {{{ an }^2}x{kern 1pt} { ext{d}}x} hfill \ = int {{{sec }^2}x{kern 1pt} { ext{d}}x} - int {1{kern 1pt} { ext{d}}x} hfill \ = an x - x + C hfill \ intlimits_0^{frac{pi }{4}} {{{ an }^2}} left( x ight){kern 1pt} { ext{d}}x = 1 - frac{pi }{4} hfill \ end{gathered} ]$

217.

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

218.

$[egin{gathered} int {frac{{cos x{{ ext{e}}^{ln left| {{x^2} - 1} ight|}}}}{{{x^2} - 1}}{kern 1pt} { ext{d}}x} hfill \ = int {frac{{cos xleft| {{x^2} - 1} ight|}}{{{x^2} - 1}}{kern 1pt} { ext{d}}x} hfill \ u = frac{{left| {{x^2} - 1} ight|}}{{{x^2} - 1}},v = cos x, hfill \ u = 0,v = sin x, hfill \ = frac{{sin xleft| {{x^2} - 1} ight|}}{{{x^2} - 1}} - int {0{kern 1pt} { ext{d}}x} hfill \ = frac{{sin xleft| {{x^2} - 1} ight|}}{{{x^2} - 1}} + C hfill \ end{gathered} ]$

219.

$[egin{gathered} int {frac{1}{{{x^2} + {a^2}}}{kern 1pt} { ext{d}}x} hfill \ u = frac{x}{a},frac{{{ ext{d}}u}}{{{ ext{d}}x}} = frac{1}{a}, hfill \ = frac{1}{a}int {frac{1}{{{u^2} + 1}}{kern 1pt} { ext{d}}u} hfill \ = frac{{arctan u}}{a} hfill \ = frac{{arctan frac{x}{a}}}{a} + C hfill \ intlimits_0^{sqrt 3 a} {frac{1}{{{a^2} + {x^2}}}} {kern 1pt} { ext{d}}x = frac{pi }{{3a}} hfill \ end{gathered} ]$

220.

$[egin{gathered} int {frac{{1 + x}}{{sqrt {1 - {x^2}} }}} {kern 1pt} { ext{d}}x hfill \ = int {frac{x}{{sqrt {1 - {x^2}} }}{kern 1pt} { ext{d}}x} + int {frac{1}{{sqrt {1 - {x^2}} }}{kern 1pt} { ext{d}}x} hfill \ u = 1 - {x^2},frac{{{ ext{d}}u}}{{{ ext{d}}x}} = - 2x, hfill \ = - frac{1}{2}int {frac{1}{{sqrt u }}{kern 1pt} { ext{d}}u} + arcsin x hfill \ = arcsin x - sqrt {1 - {x^2}} + C hfill \ intlimits_0^{frac{1}{2}} {frac{{1 + x}}{{sqrt {1 - {x^2}} }}} {kern 1pt} { ext{d}}x = frac{{pi - 3sqrt 3 + 6}}{6} hfill \ end{gathered} ]$

221.

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

222.

$[egin{gathered} int {frac{{{{left( {x - 1} ight)}^2}}}{{sqrt x }}{kern 1pt} { ext{d}}x} hfill \ = int {{x^{frac{3}{2}}}{kern 1pt} { ext{d}}x} - 2int {sqrt x {kern 1pt} { ext{d}}x} + int {frac{1}{{sqrt x }}{kern 1pt} { ext{d}}x} hfill \ = frac{{2{x^{frac{5}{2}}}}}{5} - frac{{4{x^{frac{3}{2}}}}}{3} + 2sqrt x + C hfill \ = frac{{2sqrt x left( {3{x^2} - 10x + 15} ight)}}{{15}} + C hfill \ end{gathered} ]$

223.

$[egin{gathered} int {frac{1}{{1 + sqrt x }}} {kern 1pt} { ext{d}}x hfill \ u = sqrt x + 1,frac{{{ ext{d}}u}}{{{ ext{d}}x}} = frac{1}{{2sqrt x }}, hfill \ = 2int {frac{{u - 1}}{u}{kern 1pt} { ext{d}}u} hfill \ = 2(int {1{kern 1pt} { ext{d}}u} - int {frac{1}{u}{kern 1pt} { ext{d}}u} ) hfill \ = 2u - 2ln u hfill \ = 2left( {sqrt x + 1} ight) - 2ln left( {sqrt x + 1} ight) + C hfill \ = 2sqrt x - 2ln left( {sqrt x + 1} ight) + C hfill \ intlimits_0^4 {frac{1}{{1 + sqrt x }}} {kern 1pt} { ext{d}}x = 4 - 2ln 3 hfill \ end{gathered} ]$

224.

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

225.

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

226.

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

227.

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

228.

$[egin{gathered} int {{x^2}ln x{kern 1pt} { ext{d}}x} hfill \ u = ln x,v = {x^2}, hfill \ u = frac{1}{x},v = frac{{{x^3}}}{3}, hfill \ = frac{{{x^3}ln x}}{3} - int {frac{{{x^2}}}{3}{kern 1pt} { ext{d}}x} hfill \ = frac{{{x^3}ln x}}{3} - frac{{{x^3}}}{9} + C hfill \ = frac{{{x^3}left( {3ln x - 1} ight)}}{9} + C hfill \ end{gathered} ]$

229.

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

230.

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

231.

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

232.

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

233.

$[egin{gathered} int {frac{{cos sqrt x }}{{sqrt x }}{kern 1pt} { ext{d}}x} hfill \ u = sqrt x ,frac{{{ ext{d}}u}}{{{ ext{d}}x}} = frac{1}{{2sqrt x }}, hfill \ = 2int {cos u{kern 1pt} { ext{d}}u} hfill \ = 2sin u hfill \ = 2sin sqrt x + C hfill \ intlimits_0^{frac{{{pi ^2}}}{4}} {frac{{cos sqrt x }}{{sqrt x }}} {kern 1pt} { ext{d}}x = 2 hfill \ end{gathered} ]$

234.

$[egin{gathered} int {frac{1}{{sqrt {2 + {x^2}} }}{kern 1pt} { ext{d}}x} hfill \ u = frac{x}{{sqrt 2 }},frac{{{ ext{d}}u}}{{{ ext{d}}x}} = frac{1}{{sqrt 2 }}, hfill \ = int {frac{1}{{sqrt {{u^2} + 1} }}{kern 1pt} { ext{d}}u} hfill \ u = an v,v = arctan u,frac{{{ ext{d}}u}}{{{ ext{d}}v}} = {sec ^2}v, hfill \ = int {frac{{{{sec }^2}v}}{{sqrt {{{ an }^2}v + 1} }}{kern 1pt} { ext{d}}v} = int {sec v{kern 1pt} { ext{d}}v} hfill \ = ln left( { an v + sec v} ight) hfill \ an left( {arctan u} ight) = u,sec left( {arctan u} ight) = sqrt {{u^2} + 1} , hfill \ = ln left( {sqrt {{u^2} + 1} + u} ight) hfill \ = ln left| {sqrt {frac{{{x^2}}}{2} + 1} + frac{x}{{sqrt 2 }}} ight| + C hfill \ = ln left| {sqrt {{x^2} + 2} + x} ight| + C hfill \ intlimits_0^{sqrt 2 } {frac{1}{{sqrt {2 + {x^2}} }}} {kern 1pt} { ext{d}}x = hfill \ end{gathered} ]$

235.

$[egin{gathered} int {frac{{sin 2x}}{{1 + {{sin }^2}x}}} {kern 1pt} { ext{d}}x hfill \ = int {cos xcdotfrac{{2sin x}}{{{{sin }^2}x + 1}}{kern 1pt} { ext{d}}x} hfill \ u = sin x,frac{{{ ext{d}}u}}{{{ ext{d}}x}} = cos x, hfill \ = 2int {frac{u}{{{u^2} + 1}}{kern 1pt} { ext{d}}u} hfill \ v = {u^2} + 1,frac{{{ ext{d}}v}}{{{ ext{d}}u}} = 2u, hfill \ = 2 imes frac{1}{2}int {frac{1}{v}{kern 1pt} { ext{d}}v} = ln v = ln left( {{u^2} + 1} ight) hfill \ = ln left( {{{sin }^2}x + 1} ight) + C hfill \ end{gathered} ]$

236.

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

237.

$[egin{gathered} int {sin } xln left( {1 + cos x} ight){kern 1pt} { ext{d}}x hfill \ u = cos x + 1,frac{{{ ext{d}}u}}{{{ ext{d}}x}} = - sin x, hfill \ = - int {ln u{kern 1pt} { ext{d}}u} hfill \ f = ln u,g = 1, hfill \ f = frac{1}{u},g = u, hfill \ = - (uln u - int {1{kern 1pt} { ext{d}}u} ) hfill \ = u - uln u hfill \ = - left( {cos x + 1} ight)ln left( {cos x + 1} ight) + cos x + 1 + C hfill \ = cos x - left( {cos x + 1} ight)ln left( {cos x + 1} ight) + C hfill \ end{gathered} ]$

238.

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

239.

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

240.

$[egin{gathered} int {frac{{sqrt {{x^2} - 1} }}{x}{kern 1pt} { ext{d}}x} hfill \ u = sqrt {{x^2} - 1} ,frac{{{ ext{d}}u}}{{{ ext{d}}x}} = frac{x}{{sqrt {{x^2} - 1} }}, hfill \ = int {frac{{{u^2}}}{{{u^2} + 1}}{kern 1pt} { ext{d}}u} hfill \ {u^2} = {u^2} + 1 - 1, hfill \ = int {1{kern 1pt} { ext{d}}u} - int {frac{1}{{{u^2} + 1}}{kern 1pt} { ext{d}}u} hfill \ = u - arctan u hfill \ = sqrt {{x^2} - 1} - arctan sqrt {{x^2} - 1} + C hfill \ intlimits_1^2 {frac{{sqrt {{x^2} - 1} }}{x}} {kern 1pt} { ext{d}}x = frac{{pi + 2cdot3sqrt 3 }}{6} - frac{pi }{2} = - frac{{pi - 3sqrt 3 }}{3} hfill \ end{gathered} ]$

241.

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

242.

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

243.

$[egin{gathered} int {frac{{3{x^4} + 3{x^2} + 1}}{{{x^2} + 1}}{kern 1pt} { ext{d}}x} hfill \ = int {frac{1}{{{x^2} + 1}}{kern 1pt} { ext{d}}x} + 3int {{x^2}{kern 1pt} { ext{d}}x} hfill \ = arctan x + {x^3} + C hfill \ end{gathered} ]$

244.

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

245.

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

246.

$[egin{gathered} int {xcos xsin x{ ext{d}}x} = frac{1}{2}int {xsin 2x{kern 1pt} { ext{d}}x} hfill \ u = { ext{x}},v = sin 2x, hfill \ u = 1,v = - frac{{cos 2x}}{2}, hfill \ = frac{1}{2}( - int { - frac{{cos 2x}}{2}{kern 1pt} { ext{d}}x} - frac{{xcos 2x}}{2}) hfill \ = frac{1}{2}(frac{{sin 2x}}{4} - frac{{xcos 2x}}{2}) hfill \ = frac{{sin 2x}}{8} - frac{{xcos 2x}}{4} + C hfill \ = frac{{sin 2x - 2xcos 2x}}{8} + C hfill \ end{gathered} ]$

247.

$[egin{gathered} int r ln left( {1 + {r^2}} ight){kern 1pt} { ext{d}}r hfill \ u = {r^2} + 1,frac{{{ ext{d}}u}}{{{ ext{d}}r}} = 2r, hfill \ = frac{1}{2}int {ln u{kern 1pt} { ext{d}}u} hfill \ f = ln u,g = 1, hfill \ f = frac{1}{u},g = u, hfill \ = frac{1}{2}(uln u - int {1{kern 1pt} { ext{d}}u} ) hfill \ = frac{{uln u}}{2} - frac{u}{2} hfill \ = frac{{left( {{r^2} + 1} ight)ln left( {{r^2} + 1} ight)}}{2} - frac{{{r^2} + 1}}{2} + C hfill \ = frac{{left( {{r^2} + 1} ight)left[ {ln left( {{r^2} + 1} ight) - 1} ight]}}{2} + C hfill \ intlimits_0^1 r ln left( {1 + {r^2}} ight){kern 1pt} { ext{d}}r = frac{{2ln 2 - 1}}{2} hfill \ end{gathered} ]$
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TAG:高等數學 | 高等數學大學課程 | 微積分 |