積分練習草稿5

02-13

149.

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

150.

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

151.

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

152.

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

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

153.

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

$[egin{gathered} = 2sqrt x - 4sqrt[4]{x} + 4ln left( {sqrt[4]{x} + 1} ight) + C hfill \ = 4left[ {frac{{sqrt x - 2sqrt[4]{x}}}{2} + ln left( {sqrt[4]{x} + 1} ight)} ight] + C hfill \ # hfill \ intlimits_0^{16} {frac{1}{{sqrt x + sqrt[4]{x}}}} {kern 1pt} { ext{d}}x = 4ln 3 hfill \ end{gathered} ]$

154.

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

155.

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

$[egin{gathered} an left( {operatorname{arc} sec u} ight) = sqrt {{u^2} - 1} ,sec left( {operatorname{arc} sec u} ight) = u, hfill \ = 2ln left( {sqrt {{u^2} - 1} + u} ight) = 2ln left( {sqrt x + sqrt {x - 1} } ight) + C hfill \ = ln left| {2sqrt {{x^2} - x} + 2x - 1} ight| + C hfill \ # hfill \ intlimits_1^{frac{3}{2}} {frac{1}{{sqrt {{x^2} - x} }}} {kern 1pt} { ext{d}}x = frac{{ln left( {4sqrt 3 + 7} ight)}}{2} hfill \ end{gathered} ]$

156.

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

157.

$[egin{gathered} int {{{cos }^2}x{{sin }^3}x{kern 1pt} { ext{d}}x} hfill \ = int { - frac{{sin 5x - 3left( {sin 3x - sin x} ight) + 2sin 3x - 5sin x}}{{16}}{kern 1pt} { ext{d}}x} hfill \ = - frac{1}{{16}}int {sin 5x{kern 1pt} { ext{d}}x} + frac{1}{{16}}int {sin 3x{ ext{d}}x} + frac{1}{8}int {sin x{kern 1pt} { ext{d}}x} hfill \ = frac{{cos 5x}}{{80}} - frac{{cos 3x}}{{48}} - frac{{cos x}}{8} + C hfill \ = frac{{3{{cos }^5}x - 5{{cos }^3}x}}{{15}} + C hfill \ = frac{{{{cos }^3}xleft( {3{{cos }^2}x - 5} ight)}}{{15}} + C hfill \ end{gathered} ]$

158.

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

159.

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

$[egin{gathered} = frac{{{pi ^2}arcsin frac{x}{pi }}}{2} + frac{{pi xsqrt {1 - frac{{{x^2}}}{{{pi ^2}}}} }}{2} + C hfill \ = frac{{{pi ^2}arcsin frac{x}{pi } + xsqrt {{pi ^2} - {x^2}} }}{2} + C hfill \ intlimits_0^pi {sqrt {{pi ^2} - {x^2}} } {kern 1pt} { ext{d}}x = frac{{{pi ^3}}}{4} hfill \ end{gathered} ]$

160.

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

161.

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

162.

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

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

163.

$[egin{gathered} int {frac{{{y^3}}}{{{{left( {1 + {x^2} + {y^4}} ight)}^2}}}{kern 1pt} { ext{d}}y} hfill \ u = {y^4} + {x^2} + 1,frac{{{ ext{d}}u}}{{{ ext{d}}y}} = 4{y^3}, hfill \ = frac{1}{4}int {frac{1}{{{u^2}}}{kern 1pt} { ext{d}}u} = - frac{1}{{4u}} hfill \ = - frac{1}{{4left( {{y^4} + {x^2} + 1} ight)}} + C hfill \ Rightarrow hfill \ int_0^{sqrt x } {frac{{{y^3}}}{{{{left( {{y^4} + {x^2} + 1} ight)}^2}}}{kern 1pt} { ext{d}}y} = - frac{1}{{4left( {2{x^2} + 1} ight)}} + frac{1}{{4left( {{x^2} + 1} ight)}} hfill \ # hfill \ int {left[ {frac{1}{{4left( {{x^2} + 1} ight)}} - frac{1}{{4left( {2{x^2} + 1} ight)}}{kern 1pt} { ext{d}}x} ight]} hfill \ = frac{1}{4}int {frac{1}{{{x^2} + 1}}{kern 1pt} { ext{d}}x} - frac{1}{4}int {frac{1}{{2{x^2} + 1}}{kern 1pt} { ext{d}}x} hfill \ end{gathered} ]$

$[egin{gathered} u = sqrt 2 x, hfill \ = frac{1}{4}arctan x - frac{1}{{sqrt 2 }}int {frac{1}{{{u^2} + 1}}{kern 1pt} { ext{d}}u} hfill \ = frac{{arctan x}}{4} - frac{{arctan sqrt 2 x}}{{{2^{frac{5}{2}}}}} + C hfill \ intlimits_0^{ + infty } {left[ { - frac{1}{{4left( {2{x^2} + 1} ight)}} + frac{1}{{4left( {{x^2} + 1} ight)}}{kern 1pt} { ext{d}}x} ight]} = frac{{left( {sqrt 2 - 1} ight)pi }}{{{2^{frac{7}{2}}}}} hfill \ Rightarrow hfill \ int_0^{ + infty } {dx} int_0^{sqrt x } {frac{{{y^3}}}{{{{left( {1 + {x^2} + {y^4}} ight)}^2}}}{kern 1pt} { ext{d}}y} = frac{{left( {sqrt 2 - 1} ight)pi }}{{{2^{frac{7}{2}}}}} hfill \ end{gathered} ]$

164.

$[egin{gathered} int {cos 3xsin 3x{kern 1pt} { ext{d}}x} hfill \ = int {frac{{sin 6x}}{2}{kern 1pt} { ext{d}}x} hfill \ { ext{u}} = { ext{6x}},frac{{{ ext{d}}u}}{{{ ext{d}}x}} = 6, hfill \ = frac{1}{{12}}int {sin u{kern 1pt} { ext{d}}u} hfill \ = - frac{{cos u}}{{12}} hfill \ = - frac{{cos 6x}}{{12}} + C hfill \ = - frac{{{{cos }^2}3x}}{6} + C hfill \ end{gathered} ]$

165.

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

166.

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

167.

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

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

$[intlimits_{sqrt 3 }^{sqrt 8 } {frac{{sqrt {{x^2} + 1} }}{x}} {kern 1pt} { ext{d}}x = frac{{ln 3 - ln 2 + 2}}{2}]$

168.

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

169.

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

170.

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

171.

$[egin{gathered} int_{ - frac{pi }{4}}^{frac{pi }{4}} {frac{{1 + an u}}{{{{sec }^2}u}}du} hfill \ = int_{ - frac{pi }{4}}^{frac{pi }{4}} {frac{{ an u}}{{{{sec }^2}u}}{kern 1pt} { ext{d}}u} + int_{ - frac{pi }{4}}^{frac{pi }{4}} {frac{1}{{{{sec }^2}u}}{kern 1pt} { ext{d}}u} hfill \ = int_{ - frac{pi }{4}}^{frac{pi }{4}} {{{cos }^2}u{kern 1pt} { ext{d}}u} = 2int_0^{frac{pi }{4}} {{{cos }^2}u{kern 1pt} { ext{d}}u} hfill \ end{gathered} ]$

172.

$[egin{gathered} int {frac{1}{{{{left( {9 - {x^2}} ight)}^{frac{3}{2}}}}}{kern 1pt} { ext{d}}x} hfill \ x = 3sin u,u = arcsin frac{x}{3},frac{{{ ext{d}}x}}{{{ ext{d}}u}} = 3cos u, hfill \ = int {frac{{3cos u}}{{{{left( {9 - 9{{sin }^2}u} ight)}^{frac{3}{2}}}}}{kern 1pt} { ext{d}}u} hfill \ = frac{1}{9}int {frac{1}{{{{cos }^2}u}}{kern 1pt} { ext{d}}u} hfill \ = frac{1}{9}int {{{sec }^2}u{kern 1pt} { ext{d}}u} = frac{1}{9} an u hfill \ an left( {arcsin frac{x}{3}} ight) = frac{x}{{3sqrt {1 - frac{{{x^2}}}{9}} }}, hfill \ = frac{x}{{27sqrt {1 - frac{{{x^2}}}{9}} }} + C hfill \ = frac{x}{{9sqrt {9 - {x^2}} }} + C hfill \ intlimits_0^{frac{3}{2}} {frac{1}{{{{left( {9 - {x^2}} ight)}^{frac{3}{2}}}}}} {kern 1pt} { ext{d}}x = frac{1}{{{3^{frac{5}{2}}}}} hfill \ end{gathered} ]$

173.

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

174.

$[egin{gathered} int {{x^3}sin {x^2}{kern 1pt} { ext{d}}x} hfill \ u = {x^2},frac{{{ ext{d}}u}}{{{ ext{d}}x}} = 2x, hfill \ = frac{1}{2}int {usin u{kern 1pt} { ext{d}}u} hfill \ f = { ext{u}},g = sin u, hfill \ f = 1,g = - cos u, hfill \ = frac{1}{2}( - int { - cos u{kern 1pt} { ext{d}}u} - ucos u) hfill \ = frac{{sin u}}{2} - frac{{ucos u}}{2} hfill \ = frac{{sin {x^2}}}{2} - frac{{{x^2}cos {x^2}}}{2} + C hfill \ = frac{{sin {x^2} - {x^2}cos {x^2}}}{2} + C hfill \ intlimits_{ - 1}^1 {{x^3}} sin {x^2}{kern 1pt} { ext{d}}x = 0 hfill \ end{gathered} ]$

175.

$[egin{gathered} int {xsqrt {1 - {x^2}} {kern 1pt} { ext{d}}x} hfill \ u = 1 - {x^2},frac{{{ ext{d}}u}}{{{ ext{d}}x}} = - 2x, hfill \ = - frac{1}{2}int {sqrt u {kern 1pt} { ext{d}}u} = - frac{{{u^{frac{3}{2}}}}}{3} hfill \ = - frac{{{{left( {1 - {x^2}} ight)}^{frac{3}{2}}}}}{3} + C hfill \ intlimits_{ - 1}^0 x sqrt {1 - {x^2}} {kern 1pt} { ext{d}}x = - frac{1}{3} hfill \ end{gathered} ]$

176.

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

177.

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

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

178.

$[egin{gathered} int {frac{{x - 5}}{{{x^3} - 3{x^2} + 4}}{kern 1pt} { ext{d}}x} hfill \ = int {frac{{x - 5}}{{{{left( {x - 2} ight)}^2}left( {x + 1} ight)}}{kern 1pt} { ext{d}}x} hfill \ frac{{x - 5}}{{{{left( {x - 2} ight)}^2}left( {x + 1} ight)}} = frac{A}{{x - 2}} + frac{B}{{{{left( {x - 2} ight)}^2}}} + frac{C}{{x + 1}}, hfill \ = int {left[ { - frac{2}{{3left( {x + 1} ight)}} + frac{2}{{3left( {x - 2} ight)}} - frac{1}{{{{left( {x - 2} ight)}^2}}}} ight]{kern 1pt} { ext{d}}x} hfill \ = - frac{2}{3}int {frac{1}{{x + 1}}{kern 1pt} { ext{d}}x} + frac{2}{3}int {frac{1}{{x - 2}}{kern 1pt} { ext{d}}x} - int {frac{1}{{{{left( {x - 2} ight)}^2}}}{kern 1pt} { ext{d}}x} hfill \ = - frac{{2ln left| {x + 1} ight|}}{3} + frac{1}{{x - 2}} + frac{{2ln left| {x - 2} ight|}}{3} + C hfill \ end{gathered} ]$

179.

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

$[egin{gathered} = 3left[ { - ln left( {u + 1} ight) + frac{{{{left( {u + 1} ight)}^3}}}{3} - frac{{3{{left( {u + 1} ight)}^2}}}{2} + 3left( {u + 1} ight)} ight] hfill \ = - 3ln left( {u + 1} ight) + {left( {u + 1} ight)^3} - frac{{9{{left( {u + 1} ight)}^2}}}{2} + 9left( {u + 1} ight) hfill \ = - 3ln left( {sqrt[6]{{{x^2} + 1}} + 1} ight) + {left( {sqrt[6]{{{x^2} + 1}} + 1} ight)^3} - frac{{9{{left( {sqrt[6]{{{x^2} + 1}} + 1} ight)}^2}}}{2} hfill \ + 9left( {sqrt[6]{{{x^2} + 1}} + 1} ight) + C hfill \ = - 3ln left( {sqrt[6]{{{x^2} + 1}} + 1} ight) + sqrt {{x^2} + 1} - frac{{3sqrt[3]{{{x^2} + 1}}}}{2} + 3sqrt[6]{{{x^2} + 1}} + C hfill \ end{gathered} ]$

180.

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

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

181.

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