積分練習草稿4

05-22

積分練習草稿4

112.

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

113.

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

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

114.

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

115.

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

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

116.

$[egin{gathered} int {{{left( {xarccos x} ight)}^2}} {kern 1pt} { ext{d}}x{ ext{ = }}int {{x^2}{{arccos }^2}x{kern 1pt} { ext{d}}x} hfill \ u = arccos x,frac{{{ ext{d}}u}}{{{ ext{d}}x}} = - frac{1}{{sqrt {1 - {x^2}} }}, hfill \ {x^2} = {cos ^2}u,{arccos ^2}x = {u^2},{cos ^2}x = 1 - {sin ^2}x, hfill \ {sin ^2}x = 1 - {cos ^2}x, - {cos ^2}x = - {cos ^2}x, hfill \ = - int {{u^2}{{cos }^2}usin u{kern 1pt} { ext{d}}u} hfill \ f = {u^2},g = {cos ^2}usin u, hfill \ f = { ext{2u}},g = - frac{{{{cos }^3}u}}{3}, hfill \ = - int {frac{{2u{{cos }^3}u}}{3}{kern 1pt} { ext{d}}u} + frac{{{u^2}{{cos }^3}u}}{3} hfill \ = - frac{2}{3}int {u{{cos }^3}u{ ext{d}}u} + frac{{{u^2}{{cos }^3}u}}{3} hfill \ = - frac{2}{3} imes int {frac{{uleft( {cos 3u + 3cos u} ight)}}{4}{kern 1pt} { ext{d}}u} + frac{{{u^2}{{cos }^3}u}}{3} hfill \ end{gathered} ]$

$[egin{gathered} = - frac{1}{6}(int {ucos 3u{kern 1pt} { ext{d}}u} + 3int {ucos u{ ext{d}}u} ) + frac{{{u^2}{{cos }^3}u}}{3} hfill \ f = { ext{u}},g = cos 3u, hfill \ f = 1,g = frac{{sin 3u}}{3}, hfill \ = - frac{1}{6}(frac{{usin 3u}}{3} - int {frac{{sin 3u}}{3}{kern 1pt} { ext{d}}u} + 3int {ucos u{kern 1pt} { ext{d}}u} ) + frac{{{u^2}{{cos }^3}u}}{3} hfill \ = - frac{1}{6}(frac{{usin 3u}}{3} + frac{{cos 3u}}{9} + 3int {ucos u{kern 1pt} { ext{d}}u} ) + frac{{{u^2}{{cos }^3}u}}{3} hfill \ f = { ext{u}},g = cos u, hfill \ f = 1,g = sin u, hfill \ = - frac{1}{6}(frac{{usin 3u}}{3} + frac{{cos 3u}}{9} + 3usin u - 3int {sin u{kern 1pt} { ext{d}}u} ) + frac{{{u^2}{{cos }^3}u}}{3} hfill \ end{gathered} ]$

$[egin{gathered} = - frac{{usin 3u}}{{18}} - frac{{cos 3u}}{{54}} - frac{1}{2}usin u - frac{1}{2}cos u{kern 1pt} + frac{{{u^2}{{cos }^3}u}}{3} hfill \ sin left( {arccos x} ight) = sqrt {1 - {x^2}} ,cos left( {arccos x} ight) = x, hfill \ = - frac{{arccos xsin left( {3arccos x} ight)}}{{18}} - frac{{cos left( {3arccos x} ight)}}{{54}} + frac{{{x^3}{{arccos }^2}x}}{3} hfill \ - frac{{sqrt {1 - {x^2}} arccos x}}{2} - frac{x}{2} hfill \ = - frac{{ - 9{x^3}{{arccos }^2}x + sqrt {1 - {x^2}} left( {6{x^2} + 12} ight)arccos x + 2{x^3} + 12x}}{{27}} + C hfill \ = frac{{9{x^3}{{arccos }^2}x + left( { - 6{x^2} - 12} ight)sqrt {1 - {x^2}} arccos x - 2{x^3} - 12x}}{{27}} + C hfill \ end{gathered} ]$

117.

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

118.

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

$[egin{gathered} = 2sqrt u ln left( {u + 1} ight) - 4v + 4arctan v hfill \ = 4arctan sqrt {{{ ext{e}}^x} - 1} + 2xsqrt {{{ ext{e}}^x} - 1} - 4sqrt {{{ ext{e}}^x} - 1} + C hfill \ = 2left( {2arctan sqrt {{{ ext{e}}^x} - 1} + left( {x - 2} ight)sqrt {{{ ext{e}}^x} - 1} } ight) + C hfill \ end{gathered} ]$

119.

$[egin{gathered} int {frac{{{{sin }^2}x}}{{{{cos }^3}x}}{kern 1pt} { ext{d}}x} = int {sec x{{ an }^2}x{kern 1pt} { ext{d}}x} hfill \ = int {{{sec }^3}x{kern 1pt} { ext{d}}x} - int {sec x{ ext{d}}x} hfill \ = frac{1}{2}smallint sec x{kern 1pt} { ext{d}}x + frac{{sec x an x}}{2} - int {sec x{kern 1pt} { ext{d}}x} hfill \ { ext{ = }}frac{{sec x an x}}{2} - frac{1}{2}int {sec x{kern 1pt} { ext{d}}x} hfill \ = frac{{sec x an x}}{2} - frac{{ln left| { an x + sec x} ight|}}{2} + C hfill \ end{gathered} ]$

120.

$[egin{gathered} int {frac{1}{{sqrt x - 3sqrt[3]{x}}}{kern 1pt} { ext{d}}x} = int {frac{1}{{6{x^{frac{5}{6}}}}}cdotfrac{{6sqrt x }}{{(sqrt[6]{x} - 3)}}{kern 1pt} { ext{d}}x} hfill \ u = sqrt[6]{x},frac{{{ ext{d}}u}}{{{ ext{d}}x}} = frac{1}{{6{x^{frac{5}{6}}}}},sqrt[3]{x} = {u^2},sqrt x = {u^3} hfill \ {x^{frac{5}{6}}} = {u^5}, hfill \ = 6int {frac{{{u^3}}}{{u - 3}}{kern 1pt} { ext{d}}u} hfill \ { ext{v}} = { ext{u}} - { ext{3}},frac{{{ ext{d}}v}}{{{ ext{d}}u}} = 1,{u^3} = {left( {v + 3} ight)^3}, hfill \ = 6int {frac{{{{left( {v + 3} ight)}^3}}}{v}{kern 1pt} { ext{d}}v} hfill \ = 6(int {{v^2}{kern 1pt} { ext{d}}v} + 9int {v{kern 1pt} { ext{d}}v} + 27int {frac{1}{v}{kern 1pt} { ext{d}}v} + 27int {1{kern 1pt} { ext{d}}v} ) hfill \ = 6(27ln v + frac{{{v^3}}}{3} + frac{{9{v^2}}}{2} + 27v) hfill \ end{gathered} ]$

$[egin{gathered} = 6left[ {27left( {u - 3} ight) + frac{{{{left( {u - 3} ight)}^3}}}{3} + frac{{9{{left( {u - 3} ight)}^2}}}{2} + 27ln left( {u - 3} ight)} ight] hfill \ = 2sqrt x + 9sqrt[3]{x} + 54sqrt[6]{x} + 162ln left| {sqrt[6]{x} - 3} ight| + C hfill \ end{gathered} ]$

121.

$[egin{gathered} int {frac{{arcsin {{ ext{e}}^x}}}{{{{ ext{e}}^x}}}} {kern 1pt} { ext{d}}x = int {{{ ext{e}}^{ - x}}arcsin {{ ext{e}}^x}{kern 1pt} { ext{d}}x} hfill \ { ext{u}} = - { ext{x}}, hfill \ = - int {{{ ext{e}}^u}operatorname{arccsc} {{ ext{e}}^u}{kern 1pt} { ext{d}}u} hfill \ v = {{ ext{e}}^u},frac{{{ ext{d}}v}}{{{ ext{d}}u}} = {{ ext{e}}^u}, hfill \ = - int {operatorname{arc} csc v{kern 1pt} { ext{d}}v} hfill \ f = operatorname{arc} csc v,g = 1, hfill \ f = - frac{1}{{vsqrt {{v^2} - 1} }},g = v, hfill \ = - voperatorname{arc} csc v - int {frac{1}{{sqrt {{v^2} - 1} }}{kern 1pt} { ext{d}}v} hfill \ v = sec u,u = operatorname{arc} sec v,frac{{{ ext{d}}v}}{{{ ext{d}}u}} = sec u an u, hfill \ = - voperatorname{arc} csc v - int {frac{{sec u an u}}{{sqrt {{{sec }^2}u - 1} }}{kern 1pt} { ext{d}}u} hfill \ end{gathered} ]$

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

$[egin{gathered} = - ln left( {sqrt {{{ ext{e}}^{2u}} - 1} + {{ ext{e}}^u}} ight) - {{ ext{e}}^u}operatorname{arccsc} {{ ext{e}}^u} hfill \ = - {{ ext{e}}^{ - x}}arcsin {{ ext{e}}^x} - ln left( {{{ ext{e}}^{ - x}} + sqrt {{{ ext{e}}^{ - 2x}} - 1} } ight) + C hfill \ end{gathered} ]$

122.

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

123.

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

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

124.

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

$[egin{gathered} { ext{u}} = { ext{x}} - { ext{1}}, hfill \ = frac{{{{left( {{x^2} - 2x - 1} ight)}^{frac{3}{2}}}}}{3} + 2int {sqrt {{u^2} - 2} {kern 1pt} { ext{d}}u} hfill \ u = sqrt 2 sec v,v = operatorname{arcsec} frac{u}{{sqrt 2 }},frac{{{ ext{d}}u}}{{{ ext{d}}v}} = sqrt 2 sec v an v, hfill \ = frac{{{{left( {{x^2} - 2x - 1} ight)}^{frac{3}{2}}}}}{3} + 2int {sqrt 2 sec vsqrt {2{{sec }^2}v - 2} an v{kern 1pt} { ext{d}}v} hfill \ = frac{{{{left( {{x^2} - 2x - 1} ight)}^{frac{3}{2}}}}}{3} + 4int {sec v{{ an }^2}v{kern 1pt} { ext{d}}v} hfill \ = frac{{{{left( {{x^2} - 2x - 1} ight)}^{frac{3}{2}}}}}{3} + 4(int {{{sec }^3}v{kern 1pt} { ext{d}}v} - int {sec v{kern 1pt} { ext{d}}v} ) hfill \ end{gathered} ]$

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

$[egin{gathered} = frac{{{{left( {{x^2} - 2x - 1} ight)}^{frac{3}{2}}}}}{3} + frac{{2sqrt {frac{{{{left( {x - 1} ight)}^2}}}{2} - 1} left( {x - 1} ight)}}{{sqrt 2 }} - 2ln left( {frac{{x - 1}}{{sqrt 2 }} + sqrt {frac{{{{left( {x - 1} ight)}^2}}}{2} - 1} } ight) hfill \ = frac{{sqrt {{x^2} - 2x - 1} left( {{x^2} + x - 4} ight) - 6ln left| {sqrt {{x^2} - 2x - 1} + x - 1} ight|}}{3} + C hfill \ end{gathered} ]$

125.

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

126.

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

127.

$[egin{gathered} int {{{sin }^4}t{kern 1pt} { ext{d}}t} hfill \ = int {frac{{cos 4t - 4cos 2t + 3}}{8}{kern 1pt} { ext{d}}t} hfill \ = frac{1}{8}int {cos 4t{kern 1pt} { ext{d}}t} - frac{1}{2}int {cos 2t{kern 1pt} { ext{d}}t} + frac{3}{8}int {1{kern 1pt} { ext{d}}t} hfill \ = frac{{sin 4t}}{{32}} - frac{{sin 2t}}{4} + frac{{3t}}{8} + C hfill \ = frac{{sin 4t - 8sin 2t + 12t}}{{32}} + C hfill \ end{gathered} ]$

128.

$[egin{gathered} int {{{sin }^6}t{kern 1pt} { ext{d}}t} hfill \ = int {frac{{ - cos 6t + 6cos 4t - 15cos 2t + 10}}{{32}}{kern 1pt} { ext{d}}t} hfill \ = - frac{1}{{32}}int {cos 6t{kern 1pt} { ext{d}}t} + frac{3}{{16}}int {cos 4t{kern 1pt} { ext{d}}t} - frac{{15}}{{32}}int {cos 2t{kern 1pt} { ext{d}}t} + frac{5}{{16}}int {1{kern 1pt} { ext{d}}t} hfill \ = - frac{{sin 6t}}{{192}} + frac{{3sin 4t}}{{64}} - frac{{15sin 2t}}{{64}} + frac{{5t}}{{16}} + C hfill \ = - frac{{sin 6t - 9sin 4t + 45sin 2t - 60t}}{{192}} + C hfill \ end{gathered} ]$

129.

$[egin{gathered} int {arctan sqrt x {kern 1pt} { ext{d}}x} hfill \ u = arctan sqrt x ,v = 1, hfill \ u = frac{1}{{2sqrt x left( {x + 1} ight)}},v = x, hfill \ = xarctan sqrt x - int {frac{{sqrt x }}{{2left( {x + 1} ight)}}{kern 1pt} { ext{d}}x} hfill \ u = sqrt x ,frac{{{ ext{d}}u}}{{{ ext{d}}x}} = frac{1}{{2sqrt x }},x = {u^2}, hfill \ = xarctan sqrt x - frac{1}{2} imes 2int {frac{{{u^2}}}{{{u^2} + 1}}{kern 1pt} { ext{d}}u} hfill \ = xarctan sqrt x - (int {1{kern 1pt} { ext{d}}u} - int {frac{1}{{{u^2} + 1}}{kern 1pt} { ext{d}}u} ) hfill \ = xarctan sqrt x - (u - arctan u) hfill \ = xarctan sqrt x - sqrt x + arctan sqrt x + C hfill \ = - sqrt x + left( {x + 1} ight)arctan sqrt x + C hfill \ intlimits_1^3 {arctan } sqrt x {kern 1pt} { ext{d}}x = frac{{4pi - 3sqrt 3 }}{3} - frac{{pi - 2}}{2} hfill \ end{gathered} ]$

130.

$[egin{gathered} int {frac{{{{cos }^2}t}}{{{{sin }^3}t}}{kern 1pt} { ext{d}}t} = int {{{cot }^2}tcsc t{kern 1pt} { ext{d}}t} hfill \ = int {{{csc }^3}t{kern 1pt} { ext{d}}t} - int {csc t{kern 1pt} { ext{d}}t} hfill \ = frac{1}{2}int {csc t{kern 1pt} { ext{d}}t} - frac{{cot tcsc t}}{2} - int {csc t{kern 1pt} { ext{d}}t} hfill \ = - frac{1}{2}int {csc t{kern 1pt} { ext{d}}t} - frac{{cot tcsc t}}{2} hfill \ = - frac{1}{2}( - ln left( {csc t + cot t} ight)) - frac{{cot tcsc t}}{2} hfill \ = frac{{ln left| {csc t + cot t} ight|}}{2} - frac{{cot tcsc t}}{2} + C hfill \ end{gathered} ]$

131.

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

132.

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

133.

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

$[intlimits_{ - frac{pi }{2}}^{frac{pi }{2}} {left( {{x^2} + cos x} ight)} sin x{kern 1pt} { ext{d}}x = 0]$

134.

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

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

$[ = ln left( {sqrt {{{ ext{e}}^{2x}} - 1} + {{ ext{e}}^x}} ight) + arcsin {{ ext{e}}^{ - x}} + C]$

135.

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

136.

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

137.

$[egin{gathered} int {{{sec }^3}xdx} hfill \ operatorname{u} = sec x,operatorname{dv} = {sec ^2}xdx, hfill \ operatorname{du} = an xsec xdx,operatorname{v} = int {{{sec }^2}xdx} = an x, hfill \ int {{{sec }^3}xdx} = an xsec x - int {{{ an }^2}xsec xdx} hfill \ = an xsec x - int {{{sec }^3}xdx} + int {sec xdx} hfill \ Rightarrow int {{{sec }^3}xdx} = frac{1}{2} an xsec x + frac{1}{2}int {sec xdx} hfill \ end{gathered} ]$

138.

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

139.

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

140.

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

141.

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

$[egin{gathered} = 4ln left( {sqrt {x + 1} + 1} ight) - 4left( {sqrt {x + 1} + 1} ight) + x + 1 + C hfill \ = 4ln left( {sqrt {x + 1} + 1} ight) - 4sqrt {x + 1} + x + C hfill \ end{gathered} ]$

142.

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

$[egin{gathered} = 4left[ {ln left( {u + 1} ight) + frac{{{{left( {u + 1} ight)}^2}}}{2} - 2left( {u + 1} ight)} ight] hfill \ = - 8left( {sqrt[4]{x} + 1} ight) + 2{left( {sqrt[4]{x} + 1} ight)^2} + 4ln left( {sqrt[4]{x} + 1} ight) + C hfill \ = 2sqrt x - 4sqrt[4]{x} + 4ln left( {sqrt[4]{x} + 1} ight) + C hfill \ end{gathered} ]$

143.

$[egin{gathered} int {frac{1}{{3 + {{sin }^2}v}}} {kern 1pt} { ext{d}}x hfill \ sin v = frac{{ an v}}{{sec v}},{sec ^2}v = { an ^2}v + 1, hfill \ = int {{{sec }^2}vcdotfrac{1}{{4{{ an }^2}v + 3}}{kern 1pt} { ext{d}}x} hfill \ u = an v,frac{{{ ext{d}}u}}{{{ ext{d}}x}} = {sec ^2}v, hfill \ = int {frac{1}{{4{u^2} + 3}}{kern 1pt} { ext{d}}u} hfill \ v = frac{{2u}}{{sqrt 3 }},frac{{{ ext{d}}v}}{{{ ext{d}}u}} = frac{2}{{sqrt 3 }}, hfill \ = frac{1}{{2sqrt 3 }}int {frac{1}{{{v^2} + 1}}{kern 1pt} { ext{d}}v} hfill \ = frac{{arctan v}}{{2sqrt 3 }} hfill \ = frac{{arctan frac{{2 an x}}{{sqrt 3 }}}}{{2sqrt 3 }} + C hfill \ end{gathered} ]$

144.

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

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

145.

$[egin{gathered} int {frac{1}{{{{left( {{x^2} + 1} ight)}^2}}}dx} = ? hfill \ # # hfill \ int {frac{1}{{{x^2} + 1}}dx} hfill \ operatorname{u} = frac{1}{{{x^2} + 1}},operatorname{dv} = dx,operatorname{du} = - frac{{2x}}{{{{left( {{x^2} + 1} ight)}^2}}}dx,v = x, hfill \ = frac{x}{{{x^2} + 1}} - int { - frac{{2{x^2}}}{{{{left( {{x^2} + 1} ight)}^2}}}dx} hfill \ = frac{x}{{{x^2} + 1}} + 2int {left[ { - frac{1}{{{{left( {{x^2} + 1} ight)}^2}}} + frac{{{x^2} + 1}}{{{{left( {{x^2} + 1} ight)}^2}}}} ight]dx} hfill \ = frac{x}{{{x^2} + 1}} - 2int {frac{1}{{{{left( {{x^2} + 1} ight)}^2}}}dx} + 2int {frac{1}{{{x^2} + 1}}dx} hfill \ Rightarrow int {frac{1}{{{{left( {{x^2} + 1} ight)}^2}}}dx} = frac{x}{{2left( {{x^2} + 1} ight)}} + frac{1}{2}int {frac{1}{{{x^2} + 1}}dx} hfill \ end{gathered} ]$

146.

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

147.

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

$[egin{gathered} = - frac{{xsqrt {4 - {x^2}} }}{2} + frac{{{x^2}}}{2} - 2arcsin frac{x}{2} + C hfill \ intlimits_{ - 2}^2 {(x - sqrt {4 - {x^2}} )} {kern 1pt} { ext{d}}x = - 2pi hfill \ end{gathered} ]$

148.

$[egin{gathered} int {sqrt {{x^4} + {x^3}} {kern 1pt} { ext{d}}x} = int {{x^{frac{3}{2}}}sqrt {x + 1} {kern 1pt} { ext{d}}x} hfill \ u = sqrt x ,frac{{{ ext{d}}u}}{{{ ext{d}}x}} = frac{1}{{2sqrt x }},{x^2} = {u^4},x = {u^2}, hfill \ = 2int {{u^4}sqrt {{u^2} + 1} {kern 1pt} { ext{d}}u} hfill \ u = an v,v = arctan u,frac{{{ ext{d}}u}}{{{ ext{d}}v}} = {sec ^2}v, hfill \ = 2int {{{sec }^2}v{{ an }^4}vsqrt {{{ an }^2}v + 1} {kern 1pt} { ext{d}}v} hfill \ = 2int {{{sec }^3}v{{ an }^4}v{kern 1pt} { ext{d}}v} hfill \ = 2int {{{sec }^7}v{kern 1pt} { ext{d}}v} - 4int {{{sec }^5}v{ ext{d}}v} + 2int {{{sec }^3}v{kern 1pt} { ext{d}}v} hfill \ { ext{# }}int {{{sec }^n}v{kern 1pt} { ext{d}}v} = frac{{n - 2}}{{n - 1}}int {{{sec }^{n - 2}}v{kern 1pt} { ext{d}}v} + frac{{{{sec }^{n - 2}}v an v}}{{n - 1}}, hfill \ end{gathered} ]$

$[egin{gathered} = frac{{ln left( { an v + sec v} ight)}}{8} + frac{{{{sec }^5}v an v}}{3} - frac{{7{{sec }^3}v an v}}{{12}} + frac{{sec v an v}}{8} hfill \ an left( {arctan u} ight) = u,sec left( {arctan u} ight) = sqrt {{u^2} + 1} , hfill \ = frac{{ln left( {sqrt {{u^2} + 1} + u} ight)}}{8} + frac{{u{{left( {{u^2} + 1} ight)}^{frac{5}{2}}}}}{3} - frac{{7u{{left( {{u^2} + 1} ight)}^{frac{3}{2}}}}}{{12}} + frac{{usqrt {{u^2} + 1} }}{8} hfill \ = frac{{ln left( {sqrt {x + 1} + sqrt x } ight)}}{8} + frac{{sqrt x {{left( {x + 1} ight)}^{frac{5}{2}}}}}{3} - frac{{7sqrt x {{left( {x + 1} ight)}^{frac{3}{2}}}}}{{12}} + frac{{sqrt x sqrt {x + 1} }}{8} + C hfill \ = frac{{3ln left( {sqrt {x + 1} + sqrt x } ight) + sqrt x sqrt {x + 1} left( {8{x^2} + 2x - 3} ight)}}{{24}} + C hfill \ end{gathered} ]$
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