積分練習草稿10

05-24

積分練習草稿10

309.

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

310.

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

311.

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

312.

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

313.

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

314.

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

315.

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

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

$[egin{gathered} = - frac{1}{2}{{ ext{e}}^u} + frac{1}{2}arctan {{ ext{e}}^u} - frac{{{{ ext{e}}^{ - 2x}}arctan {{ ext{e}}^x}}}{2} hfill \ arctan {{ ext{e}}^u} = arctan {{ ext{e}}^{ - x}} = operatorname{arccot} {{ ext{e}}^x}, hfill \ = - frac{{{{ ext{e}}^{ - 2x}}arctan {{ ext{e}}^x}}}{2} + frac{{operatorname{arccot} {{ ext{e}}^x}}}{2} - frac{{{{ ext{e}}^{ - x}}}}{2} + C hfill \ end{gathered} ]$

316.

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

317.

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

318.

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

319.

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

320.

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

321.

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

322.

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

323.

$[egin{gathered} int {sqrt {{{left( {1 - {x^2}} ight)}^3}} {kern 1pt} { ext{d}}x} hfill \ x = sin u,u = arcsin x,frac{{{ ext{d}}x}}{{{ ext{d}}u}} = cos u, hfill \ = int {cos u{{left( {1 - {{sin }^2}u} ight)}^{frac{3}{2}}}{kern 1pt} { ext{d}}u} hfill \ = int {{{cos }^4}u{kern 1pt} { ext{d}}u} hfill \ = int {frac{{cos 4u + 4cos 2u + 3}}{8}{kern 1pt} { ext{d}}u} hfill \ = frac{1}{8}int {cos 4u{kern 1pt} { ext{d}}u} + frac{1}{2}int {cos 2u{kern 1pt} { ext{d}}u} + frac{3}{8}int {1{kern 1pt} { ext{d}}u} hfill \ = frac{{sin 4u}}{{32}} + frac{{sin 2u}}{4} + frac{{3u}}{8} hfill \ sin 2u = sin left( {2arcsin x} ight) = 2xsqrt {1 - {x^2}} , hfill \ = frac{{sin left( {4arcsin x} ight)}}{{32}} + frac{{3arcsin x}}{8} + frac{{xsqrt {1 - {x^2}} }}{2} + C hfill \ = - frac{{sqrt {1 - {x^2}} left( {2{x^3} - 5x} ight) - 3arcsin x}}{8} + C hfill \ end{gathered} ]$

$[intlimits_0^1 {sqrt {{{left( {1 - {x^2}} ight)}^3}} } {kern 1pt} { ext{d}}x = frac{{3pi }}{{16}}]$

324.

$[egin{gathered} int {{{ ext{e}}^{ - x}}sin 2x{kern 1pt} { ext{d}}x} hfill \ u = sin 2x,v = {{ ext{e}}^{ - x}}, hfill \ u = 2cos 2x,v = - {{ ext{e}}^{ - x}}, hfill \ = - int { - 2{{ ext{e}}^{ - x}}cos 2x{kern 1pt} { ext{d}}x} - {{ ext{e}}^{ - x}}sin 2x hfill \ u = 2cos 2x,v = - {{ ext{e}}^{ - x}}, hfill \ u = - 4sin 2x,v = {{ ext{e}}^{ - x}}, hfill \ = int { - 4{{ ext{e}}^{ - x}}sin 2x{kern 1pt} { ext{d}}x} - {{ ext{e}}^{ - x}}sin 2x - 2{{ ext{e}}^{ - x}}cos 2x hfill \ = frac{{ - {{ ext{e}}^{ - x}}sin 2x - 2{{ ext{e}}^{ - x}}cos 2x}}{5} + C hfill \ end{gathered} ]$

325.

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

326.

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

327.

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

328.

$[egin{gathered} int {frac{{{{left( {ln x + 1} ight)}^{2012}}}}{x}{kern 1pt} { ext{d}}x} hfill \ u = ln x + 1,frac{{{ ext{d}}u}}{{{ ext{d}}x}} = frac{1}{x}, hfill \ = int {{u^{2012}}{kern 1pt} { ext{d}}u} hfill \ = frac{{{u^{2013}}}}{{2013}} hfill \ = frac{{{{left( {ln x + 1} ight)}^{2013}}}}{{2013}} + C hfill \ end{gathered} ]$

329.

$[egin{gathered} int {frac{1}{{9 - {x^2}}}} ln left( {frac{{3 + x}}{{3 - x}}} ight){kern 1pt} { ext{d}}x hfill \ = int {frac{{ln left( {frac{{ - x - 3}}{{x - 3}}} ight)}}{{9 - {x^2}}}{kern 1pt} { ext{d}}x} hfill \ u = ln left( {frac{{ - x - 3}}{{x - 3}}} ight),frac{{{ ext{d}}u}}{{{ ext{d}}x}} = frac{{left[ { - frac{1}{{x - 3}} - frac{{ - x - 3}}{{{{left( {x - 3} ight)}^2}}}} ight]left( {x - 3} ight)}}{{ - x - 3}}, hfill \ = frac{1}{6}int {u{kern 1pt} { ext{d}}u} = frac{{{u^2}}}{{12}} hfill \ = frac{{{{ln }^2}left( {frac{{ - x - 3}}{{x - 3}}} ight)}}{{12}} + C hfill \ end{gathered} ]$

330.

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

331.

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

332.

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

333.

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

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

334.

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

335.

$[egin{gathered} int {tsin wt{kern 1pt} { ext{d}}t} hfill \ u = { ext{t}},v = sin wt, hfill \ u = 1,v = - frac{{cos wt}}{omega }, hfill \ = - int { - frac{{cos wt}}{omega }{kern 1pt} { ext{d}}t} - frac{{tcos wt}}{omega } hfill \ { ext{u}} = omega { ext{t}}, hfill \ = frac{1}{{{omega ^2}}}int {cos u{kern 1pt} { ext{d}}u} - frac{{tcos wt}}{omega } hfill \ = frac{1}{{{omega ^2}}}sin u - frac{{tcos wt}}{omega } hfill \ = frac{{sin wt}}{{{omega ^2}}} - frac{{tcos wt}}{omega } + C hfill \ = frac{{sin wt - omega tcos wt}}{{{omega ^2}}} + C hfill \ intlimits_0^{frac{{2pi }}{omega }} t sin wt{kern 1pt} { ext{d}}t = - frac{{2pi }}{{{omega ^2}}} hfill \ end{gathered} ]$

336.

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

337.

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

338.

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

339.

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

340.

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

341.

$frac{-1}{left(u - 1 ight) left(u + 1 ight)^{2}}=frac{A}{u + 1}+frac{B}{left(u + 1 ight)^{2}}+frac{C}{u - 1}$

342.

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

343.

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

344.

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

$[egin{gathered} = frac{{{{cos }^3}t}}{3} - cos t + frac{{tsin 2t}}{2} - int {frac{{sin 2t}}{2}{kern 1pt} { ext{d}}t} - frac{{{t^2}}}{2} + int {{t^2}sin t{kern 1pt} { ext{d}}t} hfill \ = frac{{{{cos }^3}t}}{3} - cos t + frac{{tsin 2t}}{2} + frac{{cos 2t}}{4} - frac{{{t^2}}}{2} + int {{t^2}sin t{ ext{d}}t} hfill \ u = {t^2},v = sin t, hfill \ u = { ext{2t}},v = - cos t, hfill \ = frac{{{{cos }^3}t}}{3} - cos t + frac{{tsin 2t}}{2} + frac{{cos 2t}}{4} - frac{{{t^2}}}{2} - int { - 2tcos t{ ext{d}}t} - {t^2}cos t hfill \ = frac{{{{cos }^3}t}}{3} - cos t + frac{{tsin 2t}}{2} + frac{{cos 2t}}{4} - frac{{{t^2}}}{2} + 2int {tcos t{kern 1pt} { ext{d}}t} - {t^2}cos t hfill \ u = { ext{t,v}} = cos t, hfill \ u = 1,v = sin t, hfill \ end{gathered} ]$

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

345.

$[egin{gathered} int {frac{1}{{9 - {x^2}}}{kern 1pt} { ext{d}}x} = - int {frac{1}{{{x^2} - 9}}{kern 1pt} { ext{d}}x} hfill \ = - int {frac{1}{{left( {x - 3} ight)left( {x + 3} ight)}}{kern 1pt} { ext{d}}x} hfill \ frac{1}{{left( {x - 3} ight)left( {x + 3} ight)}} = frac{A}{{x - 3}} + frac{B}{{x + 3}}, hfill \ = frac{1}{6}int {frac{1}{{x - 3}}{kern 1pt} { ext{d}}x} - frac{1}{6}int {frac{1}{{x + 3}}{kern 1pt} { ext{d}}x} hfill \ = frac{{ln left( {x - 3} ight)}}{6} - frac{{ln left( {x + 3} ight)}}{6} hfill \ = frac{{ln left| {x + 3} ight| - ln left| {x - 3} ight|}}{6} + C hfill \ end{gathered} ]$

346.

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

347.

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

348.

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

349.

$[egin{gathered} int {frac{{ln x}}{x}{kern 1pt} { ext{d}}x} hfill \ u = ln x,frac{{{ ext{d}}u}}{{{ ext{d}}x}} = frac{1}{x}, hfill \ = int {u{kern 1pt} { ext{d}}u} = frac{{{u^2}}}{2} hfill \ = frac{{{{ln }^2}x}}{2} + C hfill \ intlimits_1^9 {frac{{ln x}}{x}} {kern 1pt} { ext{d}}x = frac{{{{ln }^2}9}}{2} hfill \ end{gathered} ]$
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