積分習題及詳解12

02-13

382.

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

383.

$[egin{gathered} int {frac{{{{ln }^2}x}}{x}{kern 1pt} { ext{d}}x} hfill \ u = ln x,frac{{{ ext{d}}u}}{{{ ext{d}}x}} = frac{1}{x}, hfill \ = int {{u^2}{kern 1pt} { ext{d}}u} = frac{{{u^3}}}{3} hfill \ = frac{{{{ln }^3}x}}{3} + C hfill \ intlimits_2^{{{ ext{e}}^2}} {frac{{{{ln }^2}x}}{x}} {kern 1pt} { ext{d}}x = frac{8}{3} - frac{{{{ln }^3}2}}{3} hfill \ = - frac{{{{ln }^3}2 - 8}}{3} hfill \ end{gathered} ]$

384.

$[egin{gathered} int {frac{x}{{sqrt {5 - 4x} }}{kern 1pt} { ext{d}}x} hfill \ x = - frac{1}{4}left( {5 - 4x} ight) + frac{5}{4}, hfill \ = frac{5}{4}int {frac{1}{{sqrt {5 - 4x} }}{kern 1pt} { ext{d}}x} - frac{1}{4}int {sqrt {5 - 4x} {kern 1pt} { ext{d}}x} hfill \ { ext{u}} = { ext{5}} - { ext{4x}},frac{{{ ext{d}}u}}{{{ ext{d}}x}} = - 4, hfill \ = - frac{5}{4} imes frac{1}{4}int {frac{1}{{sqrt u }}{kern 1pt} { ext{d}}u} + frac{1}{4} imes frac{1}{4}int {sqrt u {kern 1pt} { ext{d}}u} hfill \ = - frac{5}{8}sqrt u + frac{{{u^{frac{3}{2}}}}}{{24}} hfill \ = frac{{{{left( {5 - 4x} ight)}^{frac{3}{2}}}}}{{24}} - frac{{5sqrt {5 - 4x} }}{8} + C hfill \ = frac{{left( { - 4x - 10} ight)sqrt {5 - 4x} }}{{24}} + C hfill \ intlimits_{ - 1}^1 {frac{x}{{sqrt {5 - 4x} }}} {kern 1pt} { ext{d}}x = frac{1}{6} hfill \ end{gathered} ]$

385.

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

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

386.

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

387.

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

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

388.

$frac{x}{left(x - 2 ight) left(x + 1 ight)}=frac{A}{x + 1}+frac{B}{x - 2}$

389.

$[egin{gathered} int {frac{1}{{4{x^2} + 4x + 17}}{kern 1pt} { ext{d}}x} hfill \ = int {frac{1}{{{{left( {2x + 1} ight)}^2} + 16}}{kern 1pt} { ext{d}}x} hfill \ u = frac{{2x + 1}}{4},frac{{{ ext{d}}u}}{{{ ext{d}}x}} = frac{1}{2}, hfill \ = frac{1}{8}int {frac{1}{{{u^2} + 1}}{kern 1pt} { ext{d}}u} = frac{{arctan u}}{8} hfill \ = frac{{arctan frac{{2x + 1}}{4}}}{8} + C hfill \ end{gathered} ]$

390.

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

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

391.

$[egin{gathered} int {frac{{{x^2}}}{{sqrt {{a^2} - {x^2}} }}{kern 1pt} { ext{d}}x} hfill \ = {a^2}int {frac{1}{{sqrt {{a^2} - {x^2}} }}{kern 1pt} { ext{d}}x} - int {sqrt {{a^2} - {x^2}} {kern 1pt} { ext{d}}x} hfill \ { ext{# }}1x = asin uu = arcsin frac{x}{a}frac{{{ ext{d}}x}}{{{ ext{d}}u}} = acos u hfill \ { ext{# }}2v = frac{x}{a},frac{{{ ext{d}}v}}{{{ ext{d}}x}} = frac{1}{a}, hfill \ = {a^2}int {frac{1}{{sqrt {1 - {v^2}} }}{kern 1pt} { ext{d}}v} - int {acos usqrt {{a^2} - {a^2}{{sin }^2}u} {kern 1pt} { ext{d}}u} hfill \ = {a^2}arcsin v - {a^2}int {{{cos }^2}u{kern 1pt} { ext{d}}u} hfill \ = {a^2}arcsin v - {a^2}int {frac{{cos 2u + 1}}{2}{kern 1pt} { ext{d}}u} hfill \ = {a^2}arcsin v - frac{{{a^2}sin 2u}}{4} - frac{{{a^2}u}}{2} hfill \ end{gathered} ]$

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

392.

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

393.

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

394.

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

$[egin{gathered} = 2left[ { - 2ln left( {u + 1} ight) + frac{{{{left( {u + 1} ight)}^3}}}{3} - frac{{3{{left( {u + 1} ight)}^2}}}{2} + 4left( {u + 1} ight)} ight] hfill \ = - 4ln left( {u + 1} ight) + frac{{2{{left( {u + 1} ight)}^3}}}{3} - 3{left( {u + 1} ight)^2} + 8left( {u + 1} ight) hfill \ = 8left( {sqrt {x - 1} + 1} ight) + frac{{2{{left( {sqrt {x - 1} + 1} ight)}^3}}}{3} - 3{left( {sqrt {x - 1} + 1} ight)^2} - 4ln left( {sqrt {x - 1} + 1} ight) + C hfill \ = frac{{2sqrt {x - 1} left( {x + 5} ight)}}{3} - x - 4ln left( {sqrt {x - 1} + 1} ight) + C hfill \ end{gathered} ]$

395.

$[egin{gathered} int {sqrt {8 - {x^2}} {kern 1pt} { ext{d}}x} hfill \ x = 2sqrt 2 sin u,u = arcsin frac{x}{{2sqrt 2 }},frac{{{ ext{d}}x}}{{{ ext{d}}u}} = 2sqrt 2 cos u, hfill \ = int {2sqrt 2 cos usqrt {8 - 8{{sin }^2}u} {kern 1pt} { ext{d}}u} hfill \ = 8int {{{cos }^2}u{kern 1pt} { ext{d}}u} hfill \ = 8int {frac{{cos 2u + 1}}{2}{kern 1pt} { ext{d}}u} hfill \ = 8left( {frac{1}{2}int {cos 2u{kern 1pt} { ext{d}}u} + frac{1}{2}int {1{kern 1pt} { ext{d}}u} } ight) hfill \ = 8left( {frac{{sin 2u}}{4} + frac{u}{2}} ight) hfill \ = 2sin 2u + 4u hfill \ sin 2u = sin left( {2arcsin frac{x}{{2sqrt 2 }}} ight) = frac{{xsqrt {1 - frac{{{x^2}}}{8}} }}{{sqrt 2 }}, hfill \ = 4arcsin frac{x}{{2sqrt 2 }} + sqrt 2 xsqrt {1 - frac{{{x^2}}}{8}} + C hfill \ intlimits_0^2 {sqrt {8 - {x^2}} } {kern 1pt} { ext{d}}x = pi + 2 hfill \ end{gathered} ]$

396.

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

397.

$[egin{gathered} int {frac{{2cdot{3^x} - 5cdot{2^x}}}{{{3^x}}}{kern 1pt} { ext{d}}x} hfill \ = 2int {1{kern 1pt} { ext{d}}x} - 5int {frac{{{2^x}}}{{{3^x}}}{kern 1pt} { ext{d}}x} hfill \ = 2x - frac{{5cdot{2^x}}}{{{3^x}ln frac{2}{3}}} + C hfill \ end{gathered} ]$

398.

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

399.

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

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

400.

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

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

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

401.

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

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

402.

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

403.

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

404.

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

$[ = frac{{arcsin left( {2x - 1} ight)}}{8} + frac{{left( {2x - 1} ight)sqrt {1 - {{left( {2x - 1} ight)}^2}} }}{8} + C]$

405.

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

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

406.

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

407.

$[egin{gathered} int {frac{1}{{{x^2} + 2x + 2}}{kern 1pt} { ext{d}}x} hfill \ = int {frac{1}{{{{left( {x + 1} ight)}^2} + 1}}{kern 1pt} { ext{d}}x} hfill \ { ext{u}} = { ext{x}} + { ext{1}}, hfill \ = int {frac{1}{{{u^2} + 1}}{kern 1pt} { ext{d}}u} = arctan u hfill \ = arctan left( {x + 1} ight) hfill \ = arctan left( {x + 1} ight) + C hfill \ intlimits_{ - 2}^0 {frac{1}{{{x^2} + 2x + 2}}} {kern 1pt} { ext{d}}x = frac{pi }{2} hfill \ end{gathered} ]$

408.

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

409.

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

410.

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

411.

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

Elliptic integral

$[egin{gathered} = operatorname{E} left( {u{kern 1pt} |{kern 1pt} - 1} ight) hfill \ = operatorname{E} left( {arcsin x{kern 1pt} |{kern 1pt} - 1} ight) + C hfill \ end{gathered} ]$
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