積分習題及詳解13

02-13

412.

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

413.

$[egin{gathered} int {frac{{sqrt {{x^2} - 1} }}{{{x^2}}}{kern 1pt} { ext{d}}x} hfill \ u = sqrt {{x^2} - 1} ,v = frac{1}{{{x^2}}}, hfill \ u = frac{x}{{sqrt {{x^2} - 1} }},v = - frac{1}{x}, hfill \ = - int { - frac{1}{{sqrt {{x^2} - 1} }}{kern 1pt} { ext{d}}x} - frac{{sqrt {{x^2} - 1} }}{x} hfill \ x = sec u,u = operatorname{arc} sec x,frac{{{ ext{d}}x}}{{{ ext{d}}u}} = sec u an u, hfill \ = int {frac{{sec u an u}}{{sqrt {{{sec }^2}u - 1} }}{kern 1pt} { ext{d}}u} - frac{{sqrt {{x^2} - 1} }}{x} hfill \ = int {sec u{kern 1pt} { ext{d}}u} - frac{{sqrt {{x^2} - 1} }}{x} hfill \ = int {frac{{sec u an u + {{sec }^2}u}}{{ an u + sec u}}{kern 1pt} { ext{d}}u} - frac{{sqrt {{x^2} - 1} }}{x} hfill \ v = an u + sec u,frac{{{ ext{d}}v}}{{{ ext{d}}u}} = sec u an u + {sec ^2}u, hfill \ = int {frac{1}{v}{kern 1pt} { ext{d}}v} - frac{{sqrt {{x^2} - 1} }}{x} hfill \ end{gathered} ]$

$[egin{gathered} = ln v - frac{{sqrt {{x^2} - 1} }}{x} = ln left( { an u + sec u} ight) - frac{{sqrt {{x^2} - 1} }}{x} hfill \ an u = an left( {operatorname{arc} sec x} ight) = sqrt {{x^2} - 1} ,sec u = sec left( {operatorname{arc} sec x} ight) = x, hfill \ = ln left( {sqrt {{x^2} - 1} + x} ight) - frac{{sqrt {{x^2} - 1} }}{x} hfill \ = ln left| {sqrt {{x^2} - 1} + x} ight| - frac{{sqrt {{x^2} - 1} }}{x} + C hfill \ end{gathered} ]$

414.

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

415.

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

416.

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

417.

$[egin{gathered} int {frac{{{x^2}}}{{sqrt {{a^2} - {x^2}} }}{kern 1pt} { ext{d}}x} hfill \ {x^2} = {x^2} - {a^2} + {a^2}, hfill \ = {a^2}int {frac{1}{{sqrt {{a^2} - {x^2}} }}{kern 1pt} { ext{d}}x} - int {sqrt {{a^2} - {x^2}} {kern 1pt} { ext{d}}x} hfill \ x = asin u,u = arcsin frac{x}{a},frac{{{ ext{d}}x}}{{{ ext{d}}u}} = acos u, hfill \ = {a^2}int {frac{1}{{sqrt {{a^2} - {x^2}} }}{kern 1pt} { ext{d}}x} - int {acos usqrt {{a^2} - {a^2}{{sin }^2}u} {kern 1pt} { ext{d}}u} hfill \ = {a^2}int {frac{1}{{sqrt {{a^2} - {x^2}} }}{kern 1pt} { ext{d}}x} - {a^2}int {{{cos }^2}u{kern 1pt} { ext{d}}u} hfill \ = {a^2}int {frac{1}{{sqrt {{a^2} - {x^2}} }}{kern 1pt} { ext{d}}x} - {a^2}int {frac{{cos 2u + 1}}{2}{kern 1pt} { ext{d}}u} hfill \ end{gathered} ]$

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

$[egin{gathered} = {a^2}arcsin frac{x}{a} - frac{{{a^2}arcsin frac{x}{a}}}{2} - frac{{axsqrt {1 - frac{{{x^2}}}{{{a^2}}}} }}{2} hfill \ = frac{{{a^2}arcsin frac{x}{a}}}{2} - frac{{axsqrt {1 - frac{{{x^2}}}{{{a^2}}}} }}{2} + C hfill \ = frac{{{a^2}arcsin frac{x}{{left| a ight|}} - xsqrt {{a^2} - {x^2}} }}{2} + C hfill \ end{gathered} ]$

418.

$[egin{gathered} int {frac{{sqrt {1 - {x^2}} }}{{{x^2}}}{kern 1pt} { ext{d}}x} hfill \ u = sqrt {1 - {x^2}} ,v = frac{1}{{{x^2}}}, hfill \ u = - frac{x}{{sqrt {1 - {x^2}} }},v = - frac{1}{x}, hfill \ = - int {frac{1}{{sqrt {1 - {x^2}} }}{kern 1pt} { ext{d}}x} - frac{{sqrt {1 - {x^2}} }}{x} hfill \ = - arcsin x - frac{{sqrt {1 - {x^2}} }}{x} + C hfill \ intlimits_1^{frac{{sqrt 2 }}{2}} {frac{{sqrt {1 - {x^2}} }}{{{x^2}}}} {kern 1pt} { ext{d}}x = frac{pi }{2} - frac{{pi + 4}}{4} = frac{{pi - 4}}{4} hfill \ end{gathered} ]$

419.

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

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

420.

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

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

421.

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

422.

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

423.

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

424.

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

425.

$[egin{gathered} int {cos } xcos frac{x}{2}{kern 1pt} { ext{d}}x hfill \ u = frac{x}{2},frac{{{ ext{d}}u}}{{{ ext{d}}x}} = frac{1}{2}, hfill \ = 2int {cos ucos 2u{kern 1pt} { ext{d}}u} hfill \ = 2int {frac{{cos 3u + cos u}}{2}{kern 1pt} { ext{d}}u} hfill \ = 2left( {frac{{sin 3u}}{6} + frac{{sin u}}{2}} ight) hfill \ = frac{{sin 3u}}{3} + sin u hfill \ = frac{{sin frac{{3x}}{2}}}{3} + sin frac{x}{2} + C hfill \ end{gathered} ]$

426.

$[egin{gathered} int {frac{1}{{sin xcos x}}} {kern 1pt} { ext{d}}x hfill \ = int {csc xsec x{kern 1pt} { ext{d}}x} hfill \ = int {frac{{{{sec }^2}x}}{{ an x}}{kern 1pt} { ext{d}}x} hfill \ u = an x,frac{{{ ext{d}}u}}{{{ ext{d}}x}} = {sec ^2}x, hfill \ = int {frac{1}{u}{kern 1pt} { ext{d}}u} = ln u hfill \ = ln left| { an x} ight| + C hfill \ end{gathered} ]$

427.

$[egin{gathered} frac{{2{x^2} + 2x + 13}}{{left( {x - 2} ight){{left( {{x^2} + 1} ight)}^2}}} = frac{{Ax + B}}{{{x^2} + 1}} + frac{{Cx + D}}{{{{left( {{x^2} + 1} ight)}^2}}} + frac{E}{{x - 2}} hfill \ = frac{{left( {x - 2} ight)left( {{x^2} + 1} ight)left( {Ax + B} ight) + left( {x - 2} ight)left( {Cx + D} ight) + {{left( {{x^2} + 1} ight)}^2}E}}{{left( {x - 2} ight){{left( {{x^2} + 1} ight)}^2}}} hfill \ left{ {egin{array}{*{20}{l}} {A + E = 0} \ { - 2A + B = 0} \ {A - 2B + C + 2E = 2} \ { - 2A + B - 2C + D = 2} \ { - 2B - 2D + E = 13} end{array}} ight. hfill \ frac{{2{x^2} + 2x + 13}}{{left( {x - 2} ight){{left( {{x^2} + 1} ight)}^2}}} = frac{{ - x - 2}}{{{x^2} + 1}} + frac{{ - 3x - 4}}{{{{left( {{x^2} + 1} ight)}^2}}} + frac{1}{{x - 2}} hfill \ end{gathered} ]$

428.

$[egin{gathered} pi int_0^pi {frac{{sin x}}{{pi - x}}} dx - int_0^pi {xfrac{{sin x}}{{pi - x}}} dx hfill \ = int_0^pi {pi frac{{sin x}}{{pi - x}}} dx - int_0^pi {xfrac{{sin x}}{{pi - x}}} dx hfill \ = int_0^pi {(pi - x)frac{{sin x}}{{pi - x}}dx} hfill \ = int_0^pi {sin xdx} hfill \ end{gathered} ]$

429.

$[egin{gathered} int {sqrt {{a^2}{ heta ^2} + {a^2}} {kern 1pt} { ext{d}} heta } hfill \ = aint {sqrt {{ heta ^2} + 1} {kern 1pt} { ext{d}} heta } hfill \ heta = an u,u = arctan heta ,frac{{{ ext{d}} heta }}{{{ ext{d}}u}} = {sec ^2}u, hfill \ = aint {{{sec }^2}usqrt {{{ an }^2}u + 1} {kern 1pt} { ext{d}}u} hfill \ = aint {{{sec }^3}u{kern 1pt} { ext{d}}u} hfill \ = frac{1}{2}aint {sec u{kern 1pt} { ext{d}}u} + frac{{asec u an u}}{2} hfill \ = frac{{aln left( { an u + sec u} ight)}}{2} + frac{{asec u an u}}{2} hfill \ an u = an left( {arctan heta } ight) = q,sec u = sec left( {arctan heta } ight) = sqrt {{q^2} + 1} , hfill \ = frac{{aln left( {sqrt {{ heta ^2} + 1} + heta } ight)}}{2} + frac{{a heta sqrt {{ heta ^2} + 1} }}{2} hfill \ end{gathered} ]$

$[egin{gathered} = frac{{aleft( {ln left| {sqrt {{ heta ^2} + 1} + heta } ight| + heta sqrt {{ heta ^2} + 1} } ight)}}{2} + C hfill \ intlimits_0^{2pi } {sqrt {{a^2}{ heta ^2} + {a^2}} } {kern 1pt} { ext{d}} heta = hfill \ end{gathered} ]$

430.

$[egin{gathered} int {frac{{xarctan x}}{{sqrt {1 - {x^2}} }}{kern 1pt} { ext{d}}x} hfill \ u = arctan x,v = frac{x}{{sqrt {1 - {x^2}} }}, hfill \ u = frac{1}{{{x^2} + 1}},v = - sqrt {1 - {x^2}} , hfill \ = - int {frac{{sqrt {1 - {x^2}} }}{{ - {x^2} - 1}}{kern 1pt} { ext{d}}x} - sqrt {1 - {x^2}} arctan x hfill \ int {frac{{sqrt {1 - {x^2}} }}{{{x^2} + 1}}{kern 1pt} { ext{d}}x} - sqrt {1 - {x^2}} arctan x hfill \ x = sin u,u = arcsin x,frac{{{ ext{d}}x}}{{{ ext{d}}u}} = cos u, hfill \ = int {frac{{cos usqrt {1 - {{sin }^2}u} }}{{{{sin }^2}u + 1}}{kern 1pt} { ext{d}}u} - sqrt {1 - {x^2}} arctan x hfill \ = int {frac{{{{cos }^2}u}}{{{{sin }^2}u + 1}}{kern 1pt} { ext{d}}u} - sqrt {1 - {x^2}} arctan x hfill \ end{gathered} ]$

$[egin{gathered} sin u = frac{{ an u}}{{sec u}},cos u = frac{1}{{sec u}},{sec ^2}u = { an ^2}u + 1, hfill \ = int {{{sec }^2}ucdotfrac{1}{{left( {{{ an }^2}u + 1} ight)left( {2{{ an }^2}u + 1} ight)}}{kern 1pt} { ext{d}}u} - sqrt {1 - {x^2}} arctan x hfill \ v = an u,frac{{{ ext{d}}v}}{{{ ext{d}}u}} = {sec ^2}u, hfill \ = int {frac{1}{{left( {{v^2} + 1} ight)left( {2{v^2} + 1} ight)}}{kern 1pt} { ext{d}}v} - sqrt {1 - {x^2}} arctan x hfill \ frac{1}{{left( {{v^2} + 1} ight)left( {2{v^2} + 1} ight)}} = frac{{Av + B}}{{{v^2} + 1}} + frac{{Cv + D}}{{2{v^2} + 1}} hfill \ = 2int {frac{1}{{2{v^2} + 1}}{kern 1pt} { ext{d}}v} - int {frac{1}{{{v^2} + 1}}{kern 1pt} { ext{d}}v} - sqrt {1 - {x^2}} arctan x hfill \ end{gathered} ]$

$[egin{gathered} = sqrt 2 arctan sqrt 2 v - arctan v - sqrt {1 - {x^2}} arctan x hfill \ arctan v = arctan left( { an u} ight) = u, hfill \ = sqrt 2 arctan left( {sqrt 2 an u} ight) - u - sqrt {1 - {x^2}} arctan x hfill \ an u = an left( {arcsin x} ight) = frac{x}{{sqrt {1 - {x^2}} }}, hfill \ = sqrt 2 arctan frac{{sqrt 2 x}}{{sqrt {1 - {x^2}} }} - arcsin x - sqrt {1 - {x^2}} arctan x hfill \ = sqrt 2 arctan frac{{sqrt 2 x}}{{sqrt {1 - {x^2}} }} - sqrt {1 - {x^2}} arctan x - arcsin x + C hfill \ end{gathered} ]$

431.

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

432.

$[egin{gathered} int {frac{x}{{sqrt[3]{{1 - 3x}}}}{kern 1pt} { ext{d}}x} hfill \ = - int {frac{x}{{sqrt[3]{{3x - 1}}}}{kern 1pt} { ext{d}}x} hfill \ x = frac{1}{3}left( {3x - 1} ight) + frac{1}{3}, hfill \ = - int {left[ {frac{{{{left( {3x - 1} ight)}^{frac{2}{3}}}}}{3} + frac{1}{{3sqrt[3]{{3x - 1}}}}} ight]{ ext{d}}x} hfill \ = - frac{1}{3}int {{{left( {3x - 1} ight)}^{frac{2}{3}}}{kern 1pt} { ext{d}}x} - frac{1}{3}int {frac{1}{{sqrt[3]{{3x - 1}}}}{kern 1pt} { ext{d}}x} hfill \ { ext{u}} = { ext{3x}} - { ext{1}}, hfill \ = - frac{1}{3} imes frac{1}{3}int {{u^{frac{2}{3}}}{kern 1pt} { ext{d}}u} - frac{1}{3} imes frac{1}{3}int {frac{1}{{sqrt[3]{u}}}{kern 1pt} { ext{d}}u} hfill \ = - frac{{{{left( {3x - 1} ight)}^{frac{5}{3}}}}}{{15}} - frac{{{{left( {3x - 1} ight)}^{frac{2}{3}}}}}{6} + C hfill \ end{gathered} ]$

433.

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

434.

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

435.

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

436.

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

437.

$[egin{gathered} frac{{{x^3} + 4{x^2} + x}}{{{{left( {x + 2} ight)}^2}left( {{x^2} + x + 1} ight)}} = frac{A}{{x + 2}} + frac{B}{{{{left( {x + 2} ight)}^2}}} + frac{{Cx + D}}{{{x^2} + x + 1}} hfill \ = frac{{{{left( {x + 2} ight)}^2}left( {Cx + D} ight) + left( {x + 2} ight)left( {{x^2} + x + 1} ight)A + left( {{x^2} + x + 1} ight)B}}{{{{left( {x + 2} ight)}^2}left( {{x^2} + x + 1} ight)}} hfill \ {x^3} + 4{x^2} + x = {left( {x + 2} ight)^2}left( {Cx + D} ight) + left( {x + 2} ight)left( {{x^2} + x + 1} ight)A + left( {{x^2} + x + 1} ight)B hfill \ left{ {egin{array}{*{20}{l}} {A + C = 1} \ {3A + B + 4C + D = 4} \ {3A + B + 4C + 4D = 1} \ {2A + B + 4D = 0} end{array}} ight. hfill \ frac{{{x^3} + 4{x^2} + x}}{{{{left( {x + 2} ight)}^2}left( {{x^2} + x + 1} ight)}} = frac{1}{{x + 2}} + frac{2}{{{{left( {x + 2} ight)}^2}}} + frac{{ - 1}}{{{x^2} + x + 1}} hfill \ end{gathered} ]$

438.

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

439.

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

440.

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

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