解不定積分？

01-29

$eqalign{ int {frac{{1 - {x^7}}}{{xleft( {1 + {x^7}} ight)}}{kern 1pt} { ext{d}}x} cr = int {frac{{1 + {x^7} - 2{x^7}}}{{xleft( {1 + {x^7}} ight)}}{kern 1pt} { ext{d}}x} cr = int {frac{1}{x}{kern 1pt} { ext{d}}x} - 2int {frac{{{x^6}}}{{left( {1 + {x^7}} ight)}}{kern 1pt} { ext{d}}x} cr = ln |x| - frac{2}{7}int {frac{1}{{left( {1 + {x^7}} ight)}}{kern 1pt} { ext{d}}left( {1 + {x^7}} ight)} cr = ln |x| - frac{2}{7}ln left| {1 + {x^7}} ight| + C cr # cr int {frac{1}{{3 + 2cos x}}{kern 1pt} { ext{d}}x} cr = int {frac{{{{sec }^2}frac{x}{2}}}{{{{ an }^2}frac{x}{2} + 5}}{kern 1pt} { ext{d}}x} cr u = frac{{ an frac{x}{2}}}{{sqrt 5 }},{ ext{d}}x = frac{{2sqrt 5 }}{{{{sec }^2}frac{x}{2}}}{kern 1pt} { ext{d}}u, cr = frac{2}{{sqrt 5 }}int {frac{1}{{1 + {u^2}}}{kern 1pt} { ext{d}}u} cr = frac{{2arctan u}}{{sqrt 5 }} cr = frac{{2arctan left( {frac{{ an frac{x}{2}}}{{sqrt 5 }}} ight)}}{{sqrt 5 }} + C cr # cr int {frac{{cos x - sin x}}{{cos x + sin x}}{kern 1pt} { ext{d}}x} cr u = cos x + sin x,{ ext{d}}x = frac{1}{{cos x - sin x}}{kern 1pt} { ext{d}}u, cr = int {frac{1}{u}{kern 1pt} { ext{d}}u} cr = ln u cr = ln left| {cos x + sin x} ight| + C cr}$