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#3.18
![intlimits { frac{x^{2}+4}{x^{2}-4} } , dx intlimits { frac{x^{2}+4}{x^{2}-4} } , dx](https://tex.z-dn.net/?f=+intlimits+%7B+frac%7Bx%5E%7B2%7D%2B4%7D%7Bx%5E%7B2%7D-4%7D+%7D+%2C+dx+)
![frac{x^{2}+2}{x^{2}-4} = frac{x^{4}}{x^{2}-4} + frac{2}{x^{2}-4} frac{x^{2}+2}{x^{2}-4} = frac{x^{4}}{x^{2}-4} + frac{2}{x^{2}-4}](https://tex.z-dn.net/?f=+frac%7Bx%5E%7B2%7D%2B2%7D%7Bx%5E%7B2%7D-4%7D+%3D+frac%7Bx%5E%7B4%7D%7D%7Bx%5E%7B2%7D-4%7D+%2B+frac%7B2%7D%7Bx%5E%7B2%7D-4%7D+)
Далее применяем правило суммы:
![= intlimits { frac{x^{4}}{x^{2}-4} } dx+ intlimits { frac{2}{x^{2}-4} } , dx = intlimits { frac{x^{4}}{x^{2}-4} } dx+ intlimits { frac{2}{x^{2}-4} } , dx](https://tex.z-dn.net/?f=%3D+intlimits+%7B+frac%7Bx%5E%7B4%7D%7D%7Bx%5E%7B2%7D-4%7D+%7D+dx%2B+intlimits+%7B+frac%7B2%7D%7Bx%5E%7B2%7D-4%7D+%7D+%2C+dx+)
=>извлекаем константу (C)
=>применяем подстановку интеграла:
=>извлекаем константу
![=-8* intlimits { frac{u^{4}}{-u^{2}+1} } , du =-8* intlimits { frac{u^{4}}{-u^{2}+1} } , du](https://tex.z-dn.net/?f=%3D-8%2A+intlimits+%7B+frac%7Bu%5E%7B4%7D%7D%7B-u%5E%7B2%7D%2B1%7D+%7D+%2C+du)
![frac{u^{4}}{-u^{2}+1} =-u^{2}+ frac{u^{2}}{-u^{2}+1} frac{u^{4}}{-u^{2}+1} =-u^{2}+ frac{u^{2}}{-u^{2}+1}](https://tex.z-dn.net/?f=+frac%7Bu%5E%7B4%7D%7D%7B-u%5E%7B2%7D%2B1%7D+%3D-u%5E%7B2%7D%2B+frac%7Bu%5E%7B2%7D%7D%7B-u%5E%7B2%7D%2B1%7D+)
=> снова правило суммы:
![=-8(- intlimits {u^{2}du- intlimits {1} , du + intlimits { frac{1}{-u^{2}+1} du} } ) =-8(- intlimits {u^{2}du- intlimits {1} , du + intlimits { frac{1}{-u^{2}+1} du} } )](https://tex.z-dn.net/?f=%3D-8%28-+intlimits+%7Bu%5E%7B2%7Ddu-+intlimits+%7B1%7D+%2C+du+%2B+intlimits+%7B+frac%7B1%7D%7B-u%5E%7B2%7D%2B1%7D+du%7D++%7D+%29)
![intlimits {u^{2}du= frac{u^{2+1}}{2+1} = frac{u^{3}}{3} intlimits {u^{2}du= frac{u^{2+1}}{2+1} = frac{u^{3}}{3}](https://tex.z-dn.net/?f=+intlimits+%7Bu%5E%7B2%7Ddu%3D++frac%7Bu%5E%7B2%2B1%7D%7D%7B2%2B1%7D+%3D+frac%7Bu%5E%7B3%7D%7D%7B3%7D+)
![intlimits {1du}=u intlimits {1du}=u](https://tex.z-dn.net/?f=+intlimits+%7B1du%7D%3Du)
![intlimits { frac{1}{-u^{2}+1} } du= frac{ln|u+1|}{2} - frac{ln|u-1|}{2} intlimits { frac{1}{-u^{2}+1} } du= frac{ln|u+1|}{2} - frac{ln|u-1|}{2}](https://tex.z-dn.net/?f=+intlimits+%7B+frac%7B1%7D%7B-u%5E%7B2%7D%2B1%7D+%7D+du%3D+frac%7Bln%7Cu%2B1%7C%7D%7B2%7D+-+frac%7Bln%7Cu-1%7C%7D%7B2%7D+)
=> делаем обратную замену u=x/2:
![=-8( -frac{( frac{x}{2} )^{3}}{3}- frac{x}{2}+ frac{ln| frac{x}{2}+1 |}{2} - frac{ln| frac{x}{2}-1 |}{2} ) =-8( -frac{( frac{x}{2} )^{3}}{3}- frac{x}{2}+ frac{ln| frac{x}{2}+1 |}{2} - frac{ln| frac{x}{2}-1 |}{2} )](https://tex.z-dn.net/?f=%3D-8%28+-frac%7B%28+frac%7Bx%7D%7B2%7D+%29%5E%7B3%7D%7D%7B3%7D-+frac%7Bx%7D%7B2%7D%2B+frac%7Bln%7C+frac%7Bx%7D%7B2%7D%2B1+%7C%7D%7B2%7D+-+frac%7Bln%7C+frac%7Bx%7D%7B2%7D-1+%7C%7D%7B2%7D+++%29)
![frac{ (frac{x}{2})^{3} }{3}= frac{x^{3}}{2^{3}*3} frac{ (frac{x}{2})^{3} }{3}= frac{x^{3}}{2^{3}*3}](https://tex.z-dn.net/?f=+frac%7B+%28frac%7Bx%7D%7B2%7D%29%5E%7B3%7D+%7D%7B3%7D%3D+frac%7Bx%5E%7B3%7D%7D%7B2%5E%7B3%7D%2A3%7D+)
![=-8(- frac{x^{3}}{2^{3}*3}- frac{x}{2}+ frac{ln| frac{x}{2}+1 |}{2} - frac{ln| frac{x}{2}-1 |}{2} ) =-8(- frac{x^{3}}{2^{3}*3}- frac{x}{2}+ frac{ln| frac{x}{2}+1 |}{2} - frac{ln| frac{x}{2}-1 |}{2} )](https://tex.z-dn.net/?f=%3D-8%28-+frac%7Bx%5E%7B3%7D%7D%7B2%5E%7B3%7D%2A3%7D-+frac%7Bx%7D%7B2%7D%2B+frac%7Bln%7C+frac%7Bx%7D%7B2%7D%2B1+%7C%7D%7B2%7D+-+frac%7Bln%7C+frac%7Bx%7D%7B2%7D-1+%7C%7D%7B2%7D+++%29)
![2^{3}*3=24 2^{3}*3=24](https://tex.z-dn.net/?f=2%5E%7B3%7D%2A3%3D24)
![=-8(- frac{x^{3}}{24}- frac{x}{2}+ frac{ln| frac{x}{2}+1 |}{2} - frac{ln| frac{x}{2}-1 |}{2} ) =-8(- frac{x^{3}}{24}- frac{x}{2}+ frac{ln| frac{x}{2}+1 |}{2} - frac{ln| frac{x}{2}-1 |}{2} )](https://tex.z-dn.net/?f=%3D-8%28-+frac%7Bx%5E%7B3%7D%7D%7B24%7D-+frac%7Bx%7D%7B2%7D%2B+frac%7Bln%7C+frac%7Bx%7D%7B2%7D%2B1+%7C%7D%7B2%7D+-+frac%7Bln%7C+frac%7Bx%7D%7B2%7D-1+%7C%7D%7B2%7D+++%29)
![=-8 (-frac{x^{3}}{24}- frac{x}{2}+ frac{1}{2}ln| frac{x}{y} +1|- frac{1}{2} ln | frac{x}{2} -1|) =-8 (-frac{x^{3}}{24}- frac{x}{2}+ frac{1}{2}ln| frac{x}{y} +1|- frac{1}{2} ln | frac{x}{2} -1|)](https://tex.z-dn.net/?f=%3D-8+%28-frac%7Bx%5E%7B3%7D%7D%7B24%7D-+frac%7Bx%7D%7B2%7D%2B+frac%7B1%7D%7B2%7Dln%7C+frac%7Bx%7D%7By%7D+%2B1%7C-+frac%7B1%7D%7B2%7D+ln+%7C++++frac%7Bx%7D%7B2%7D+-1%7C%29)
=> применяем подстановку x=2u:
![=2* intlimits frac{1}{2(u^{2}-1)}du=2* frac{1}{2}* intlimits frac{1}{u^{2}-1}du =2* intlimits frac{1}{2(u^{2}-1)}du=2* frac{1}{2}* intlimits frac{1}{u^{2}-1}du](https://tex.z-dn.net/?f=%3D2%2A+intlimits++frac%7B1%7D%7B2%28u%5E%7B2%7D-1%29%7Ddu%3D2%2A+frac%7B1%7D%7B2%7D%2A+intlimits++frac%7B1%7D%7Bu%5E%7B2%7D-1%7Ddu++)
![=2* frac{1}{2} * intlimits frac{1}{-(-u^{2}+1)}du=2* frac{1}{2} (- intlimits frac{1}{-u^{2}+1} =2* frac{1}{2} * intlimits frac{1}{-(-u^{2}+1)}du=2* frac{1}{2} (- intlimits frac{1}{-u^{2}+1}](https://tex.z-dn.net/?f=%3D2%2A+frac%7B1%7D%7B2%7D+%2A+intlimits++frac%7B1%7D%7B-%28-u%5E%7B2%7D%2B1%29%7Ddu%3D2%2A+frac%7B1%7D%7B2%7D+%28-+intlimits+frac%7B1%7D%7B-u%5E%7B2%7D%2B1%7D+)
=> обратная замена u=x/2:
=> упрощаем:
![=2* frac{1}{2} ( frac{ln| frac{x}{2}+1 |}{2}- frac{ln| frac{x}{2} |-1}{2} ) =2* frac{1}{2} ( frac{ln| frac{x}{2}+1 |}{2}- frac{ln| frac{x}{2} |-1}{2} )](https://tex.z-dn.net/?f=%3D2%2A+frac%7B1%7D%7B2%7D+%28+frac%7Bln%7C+frac%7Bx%7D%7B2%7D%2B1+%7C%7D%7B2%7D-+frac%7Bln%7C+frac%7Bx%7D%7B2%7D+%7C-1%7D%7B2%7D++%29)
![= frac{1*2( frac{ln| frac{x}{2}+1 |}{2}- frac{ln |frac{x}{2} -1|}{2} )}{2} = frac{1*2( frac{ln| frac{x}{2}+1 |}{2}- frac{ln |frac{x}{2} -1|}{2} )}{2}](https://tex.z-dn.net/?f=%3D++frac%7B1%2A2%28+frac%7Bln%7C+frac%7Bx%7D%7B2%7D%2B1+%7C%7D%7B2%7D-+frac%7Bln+%7Cfrac%7Bx%7D%7B2%7D+-1%7C%7D%7B2%7D++%29%7D%7B2%7D+)
![= -frac{2 frac{ln| frac{x}{2}+1 |}{2}- frac{ln| frac{x}{2}-1 |}{2} }{2} = -frac{2 frac{ln| frac{x}{2}+1 |}{2}- frac{ln| frac{x}{2}-1 |}{2} }{2}](https://tex.z-dn.net/?f=%3D+-frac%7B2+frac%7Bln%7C+frac%7Bx%7D%7B2%7D%2B1+%7C%7D%7B2%7D-+frac%7Bln%7C+frac%7Bx%7D%7B2%7D-1+%7C%7D%7B2%7D++%7D%7B2%7D+)
![=- frac{ln | frac{x}{2}+1| }{2}- frac{ln| frac{x}{2}-1 |}{2} =- frac{ln | frac{x}{2}+1| }{2}- frac{ln| frac{x}{2}-1 |}{2}](https://tex.z-dn.net/?f=%3D-+frac%7Bln+%7C+frac%7Bx%7D%7B2%7D%2B1%7C+%7D%7B2%7D-+frac%7Bln%7C+frac%7Bx%7D%7B2%7D-1+%7C%7D%7B2%7D++)
![=- frac{ln| frac{x}{2}+1 |}{2}+ frac{ln| frac{x}{2}-1 |}{2} =- frac{ln| frac{x}{2}+1 |}{2}+ frac{ln| frac{x}{2}-1 |}{2}](https://tex.z-dn.net/?f=%3D-+frac%7Bln%7C+frac%7Bx%7D%7B2%7D%2B1+%7C%7D%7B2%7D%2B+frac%7Bln%7C+frac%7Bx%7D%7B2%7D-1+%7C%7D%7B2%7D++)
![=- frac{1}{2} ln| frac{x}{2}+1 |+ frac{1}{2} ln| frac{x}{2}-1 | =- frac{1}{2} ln| frac{x}{2}+1 |+ frac{1}{2} ln| frac{x}{2}-1 |](https://tex.z-dn.net/?f=%3D-+frac%7B1%7D%7B2%7D+ln%7C+frac%7Bx%7D%7B2%7D%2B1+%7C%2B+frac%7B1%7D%7B2%7D+ln%7C+frac%7Bx%7D%7B2%7D-1+%7C)
![=-8(- frac{x^{3}}{24} - frac{x}{2}+ frac{1}{2}ln| frac{x}{2} +1|- frac{1}{2} ln| frac{x}{2}-1 |)- frac{1}{2}ln| frac{x}{2}+1 |+ frac{1}{2}ln| frac{x}{2}-1| =-8(- frac{x^{3}}{24} - frac{x}{2}+ frac{1}{2}ln| frac{x}{2} +1|- frac{1}{2} ln| frac{x}{2}-1 |)- frac{1}{2}ln| frac{x}{2}+1 |+ frac{1}{2}ln| frac{x}{2}-1|](https://tex.z-dn.net/?f=%3D-8%28-+frac%7Bx%5E%7B3%7D%7D%7B24%7D+-+frac%7Bx%7D%7B2%7D%2B+frac%7B1%7D%7B2%7Dln%7C+frac%7Bx%7D%7B2%7D+%2B1%7C-+frac%7B1%7D%7B2%7D+ln%7C+frac%7Bx%7D%7B2%7D-1+%7C%29-+frac%7B1%7D%7B2%7Dln%7C+frac%7Bx%7D%7B2%7D%2B1+%7C%2B+frac%7B1%7D%7B2%7Dln%7C+frac%7Bx%7D%7B2%7D-1%7C)
=>упрощаем и получаем ответ:
![boxed{frac{x^{3}}{3}+4x- frac{9}{2}ln| frac{x}{2}+1 |+ frac{9}{2}ln| frac{x}{2}-1 | +C} boxed{frac{x^{3}}{3}+4x- frac{9}{2}ln| frac{x}{2}+1 |+ frac{9}{2}ln| frac{x}{2}-1 | +C}](https://tex.z-dn.net/?f=boxed%7Bfrac%7Bx%5E%7B3%7D%7D%7B3%7D%2B4x-+frac%7B9%7D%7B2%7Dln%7C+frac%7Bx%7D%7B2%7D%2B1+%7C%2B+frac%7B9%7D%7B2%7Dln%7C+frac%7Bx%7D%7B2%7D-1+%7C++%2BC%7D+)
Далее применяем правило суммы:
=>упрощаем и получаем ответ:
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