積分練習9

02-08

349.

$[egin{gathered} int {{{ ext{e}}^{sqrt[3]{x}}}{kern 1pt} { ext{d}}x} hfill \ u = sqrt[3]{x},frac{{{ ext{d}}u}}{{{ ext{d}}x}} = frac{1}{{3{x^{frac{2}{3}}}}},{x^{frac{2}{3}}} = {u^2}, hfill \ = 3int {{u^2}{{ ext{e}}^u}{kern 1pt} { ext{d}}u} hfill \ w = {u^2},g = {{ ext{e}}^u}, hfill \ w = { ext{2u}},g = {{ ext{e}}^u}, hfill \ = 3{u^2}{{ ext{e}}^u} - 6int {u{{ ext{e}}^u}{kern 1pt} { ext{d}}u} hfill \ w = u,g = {{ ext{e}}^u}, hfill \ w = { ext{1}},g = {{ ext{e}}^u}, hfill \ = 3{u^2}{{ ext{e}}^u} - 6(u{{ ext{e}}^u} - int {{{ ext{e}}^u}{kern 1pt} { ext{d}}u} ) hfill \ = 3{u^2}{{ ext{e}}^u} - 6u{{ ext{e}}^u} + 6int {{{ ext{e}}^u}{kern 1pt} { ext{d}}u} hfill \ end{gathered} ]$

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

350.

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

351.

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

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

352.

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

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

353.

$[egin{gathered} int {frac{1}{{{x^2} + 16x + 80}}{kern 1pt} { ext{d}}x} = int {frac{1}{{{{left( {x + 8} ight)}^2} + 16}}{kern 1pt} { ext{d}}x} hfill \ u = frac{{x + 8}}{4},frac{{{ ext{d}}u}}{{{ ext{d}}x}} = frac{1}{4}, hfill \ = frac{1}{4}int {frac{1}{{{u^2} + 1}}{kern 1pt} { ext{d}}u} = frac{{arctan u}}{4} hfill \ = frac{{arctan frac{{2x + 16}}{8}}}{4} + C hfill \ end{gathered} ]$

354.

$[egin{gathered} int {{{cot }^6}x{{sec }^3}x{kern 1pt} { ext{d}}x} hfill \ sec x = frac{{csc x}}{{cot x}},{cot ^2}x = {csc ^2}x - 1, hfill \ = int { - cot xcsc xcdotleft[ { - {{csc }^2}xleft( {{{csc }^2}x - 1} ight)} ight]{kern 1pt} { ext{d}}x} hfill \ u = csc x,frac{{{ ext{d}}u}}{{{ ext{d}}x}} = - cot xcsc x, hfill \ = - int {{u^2}left( {{u^2} - 1} ight){kern 1pt} { ext{d}}u} = - int {{u^4}{kern 1pt} { ext{d}}u} + int {{u^2}{kern 1pt} { ext{d}}u} hfill \ = frac{{{u^3}}}{3} - frac{{{u^5}}}{5} hfill \ = frac{{{{csc }^3}x}}{3} - frac{{{{csc }^5}x}}{5} + C hfill \ = frac{{5{{sin }^2}x - 3}}{{15{{sin }^5}x}} + C hfill \ end{gathered} ]$

355.

$[egin{gathered} int {frac{{{x^3}}}{{x + 3}}{kern 1pt} { ext{d}}x} hfill \ { ext{u}} = { ext{x}} + { ext{3}},frac{{{ ext{d}}u}}{{{ ext{d}}x}} = 1,{x^3} = {left( {u - 3} ight)^3}, hfill \ = int {frac{{{{left( {u - 3} ight)}^3}}}{u}{kern 1pt} { ext{d}}u} hfill \ = int {{u^2}{kern 1pt} { ext{d}}u} - 9int {u{kern 1pt} { ext{d}}u} - 27int {frac{1}{u}{kern 1pt} { ext{d}}u} + 27int {1{kern 1pt} { ext{d}}u} hfill \ = - 27ln u + frac{{{u^3}}}{3} - frac{{9{u^2}}}{2} + 27u hfill \ = frac{{xleft( {2{x^2} - 9x + 54} ight)}}{6} - 27ln left| {x + 3} ight| + C hfill \ end{gathered} ]$

356.

$[egin{gathered} int {xcos xsin x{kern 1pt} { ext{d}}x} hfill \ = int {frac{{xsin 2x}}{2}{kern 1pt} { ext{d}}x} = frac{1}{2}int {xsin 2x{kern 1pt} { ext{d}}x} hfill \ u = x,v = sin 2x, hfill \ u = 1,v = - frac{{cos 2x}}{2}, hfill \ = frac{1}{2}( - int { - frac{{cos 2x}}{2}{kern 1pt} { ext{d}}x} - frac{{xcos 2x}}{2}) hfill \ = frac{{sin 2x}}{8} - frac{{xcos 2x}}{4} + C hfill \ end{gathered} ]$

357.

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

358.

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

359.

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

360.

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

361.

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

362.

$[egin{gathered} int {xcos frac{x}{2}{kern 1pt} { ext{d}}x} hfill \ u = frac{x}{2},frac{{{ ext{d}}u}}{{{ ext{d}}x}} = frac{1}{2}, hfill \ = 4int {ucos u{ ext{d}}u} hfill \ w = u,g = cos u, hfill \ w = 1,g = sin u, hfill \ = 4(usin u - int {sin u{kern 1pt} { ext{d}}u} ) hfill \ = 4usin u + 4cos u hfill \ = 2xsin frac{x}{2} + 4cos frac{x}{2} + C hfill \ end{gathered} ]$

363.

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

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

364.

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

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

365.

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

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

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

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

366.

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

367.

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

368.

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

369.

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

370.

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

371.

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

372.

$[egin{gathered} int {frac{{1 - x}}{{sqrt {9 - 4{x^2}} }}{kern 1pt} { ext{d}}x} = - int {frac{{x - 1}}{{sqrt {9 - 4{x^2}} }}{kern 1pt} { ext{d}}x} hfill \ = - (int {frac{x}{{sqrt {9 - 4{x^2}} }}{kern 1pt} { ext{d}}x} - int {frac{1}{{sqrt {9 - 4{x^2}} }}{kern 1pt} { ext{d}}x} ) hfill \ u = 9 - 4{x^2},frac{{{ ext{d}}u}}{{{ ext{d}}x}} = - 8x, hfill \ = - ( - frac{1}{8}int {frac{1}{{sqrt u }}{kern 1pt} { ext{d}}u} - int {frac{1}{{sqrt {9 - 4{x^2}} }}{kern 1pt} { ext{d}}x} ) hfill \ = - ( - frac{{sqrt u }}{4} - int {frac{1}{{sqrt {9 - 4{x^2}} }}{kern 1pt} { ext{d}}x} ) hfill \ = - ( - frac{{sqrt {9 - 4{x^2}} }}{4} - int {frac{1}{{sqrt {9 - 4{x^2}} }}{kern 1pt} { ext{d}}x} ) hfill \ u = frac{{2x}}{3},frac{{{ ext{d}}u}}{{{ ext{d}}x}} = frac{2}{3}, hfill \ = - ( - frac{{sqrt {9 - 4{x^2}} }}{4} - frac{1}{2}int {frac{1}{{sqrt {1 - {u^2}} }}{kern 1pt} { ext{d}}u} ) hfill \ end{gathered} ]$

$[egin{gathered} = - ( - frac{{sqrt {9 - 4{x^2}} }}{4} - frac{{arcsin u}}{2}) hfill \ = - ( - frac{{sqrt {9 - 4{x^2}} }}{4} - frac{{arcsin frac{{2x}}{3}}}{2}) hfill \ = frac{{sqrt {9 - 4{x^2}} }}{4} + frac{{arcsin frac{{2x}}{3}}}{2} + C hfill \ end{gathered} ]$

373.

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

374.

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

375.

$[egin{gathered} int {sin 5xsin 7x{kern 1pt} { ext{d}}x} = int { - frac{{cos 12x - cos 2x}}{2}{kern 1pt} { ext{d}}x} hfill \ = frac{1}{2}int {cos 2x{kern 1pt} { ext{d}}x} - frac{1}{2}int {cos 12x{kern 1pt} { ext{d}}x} hfill \ { ext{u}} = { ext{12x}},frac{{{ ext{d}}u}}{{{ ext{d}}x}} = 12, hfill \ = frac{1}{2}int {cos 2x{kern 1pt} { ext{d}}x} - frac{1}{2} imes frac{1}{{12}}int {cos u{kern 1pt} { ext{d}}u} hfill \ = frac{1}{2}int {cos 2x{kern 1pt} { ext{d}}x} - frac{{sin 12x}}{{24}} hfill \ = frac{{sin 2x}}{4} - frac{{sin 12x}}{{24}} + C hfill \ end{gathered} ]$

376.

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

377.

$[egin{gathered} int {sin 2xcos 3x{kern 1pt} { ext{d}}x} = int {frac{{sin 5x - sin x}}{2}{kern 1pt} { ext{d}}x} hfill \ = frac{1}{2}int {sin 5x{kern 1pt} { ext{d}}x} - frac{1}{2}int {sin x{kern 1pt} { ext{d}}x} hfill \ = frac{{cos x}}{2} - frac{{cos 5x}}{{10}} + C hfill \ end{gathered} ]$

378.

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

379.

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

380.

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

381.

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