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1) Этот пример не имеет смысла решать интегрируя частями
![intlimits {sin^3(x)} , dx = intlimits {sin^2(x)} , d(-cos(x))=- intlimits {(1-cos^2(x))} , d(cos(x))= intlimits {sin^3(x)} , dx = intlimits {sin^2(x)} , d(-cos(x))=- intlimits {(1-cos^2(x))} , d(cos(x))=](https://tex.z-dn.net/?f=+intlimits+%7Bsin%5E3%28x%29%7D+%2C+dx+%3D+intlimits+%7Bsin%5E2%28x%29%7D+%2C+d%28-cos%28x%29%29%3D-+intlimits+%7B%281-cos%5E2%28x%29%29%7D+%2C+d%28cos%28x%29%29%3D++)
![=- intlimits { , d(cos(x))+ intlimits {cos^2(x)} , d(cos(x)) =-cos(x)+ frac{cos^3(x)}{3}+C =- intlimits { , d(cos(x))+ intlimits {cos^2(x)} , d(cos(x)) =-cos(x)+ frac{cos^3(x)}{3}+C](https://tex.z-dn.net/?f=%3D-+intlimits+%7B+%2C+d%28cos%28x%29%29%2B+intlimits+%7Bcos%5E2%28x%29%7D+%2C+d%28cos%28x%29%29+%3D-cos%28x%29%2B+frac%7Bcos%5E3%28x%29%7D%7B3%7D%2BC+)
2)
![intlimits { frac{ln^2(x)}{x^2} } , dx =
intlimits {ln^2(x)} , d(- frac{1}{x} ) =
ln^2(x)*(- frac{1}{x} ) - intlimits {(- frac{1}{x} )} , d(ln^2(x)) =
intlimits { frac{ln^2(x)}{x^2} } , dx =
intlimits {ln^2(x)} , d(- frac{1}{x} ) =
ln^2(x)*(- frac{1}{x} ) - intlimits {(- frac{1}{x} )} , d(ln^2(x)) =](https://tex.z-dn.net/?f=+intlimits+%7B+frac%7Bln%5E2%28x%29%7D%7Bx%5E2%7D+%7D+%2C+dx+%3D%0A+intlimits+%7Bln%5E2%28x%29%7D+%2C+d%28-+frac%7B1%7D%7Bx%7D+%29++%3D%0A+ln%5E2%28x%29%2A%28-+frac%7B1%7D%7Bx%7D+%29++-+intlimits+%7B%28-+frac%7B1%7D%7Bx%7D+%29%7D+%2C+d%28ln%5E2%28x%29%29+%3D%0A)
![= ln^2(x)*(- frac{1}{x} ) + intlimits { frac{1}{x}*2*ln(x)* frac{1}{x} } , dx = = ln^2(x)*(- frac{1}{x} ) + intlimits { frac{1}{x}*2*ln(x)* frac{1}{x} } , dx =](https://tex.z-dn.net/?f=%3D+ln%5E2%28x%29%2A%28-+frac%7B1%7D%7Bx%7D+%29++%2B+intlimits+%7B+frac%7B1%7D%7Bx%7D%2A2%2Aln%28x%29%2A+frac%7B1%7D%7Bx%7D+%7D+%2C+dx+%3D)
![= - frac{ln^2(x)}{x} +2 intlimits {ln(x)} , d(- frac{1}{x} ) = = - frac{ln^2(x)}{x} +2 intlimits {ln(x)} , d(- frac{1}{x} ) =](https://tex.z-dn.net/?f=%3D+-+frac%7Bln%5E2%28x%29%7D%7Bx%7D++%2B2+intlimits+%7Bln%28x%29%7D+%2C+d%28-+frac%7B1%7D%7Bx%7D+%29+%3D)
![= - frac{ln^2(x)}{x} +2[ln(x)*(- frac{1}{x} )- intlimits {(- frac{1}{x} )} , d(ln(x))] = = - frac{ln^2(x)}{x} +2[ln(x)*(- frac{1}{x} )- intlimits {(- frac{1}{x} )} , d(ln(x))] =](https://tex.z-dn.net/?f=%3D+-+frac%7Bln%5E2%28x%29%7D%7Bx%7D++%2B2%5Bln%28x%29%2A%28-+frac%7B1%7D%7Bx%7D+%29-+intlimits+%7B%28-+frac%7B1%7D%7Bx%7D+%29%7D+%2C+d%28ln%28x%29%29%5D++%3D)
![= - frac{ln^2(x)}{x} - frac{2ln(x)}{x}+ 2intlimits { frac{1}{x} * frac{1}{x} } , dx = = - frac{ln^2(x)}{x} - frac{2ln(x)}{x}+ 2intlimits { frac{1}{x} * frac{1}{x} } , dx =](https://tex.z-dn.net/?f=%3D+-+frac%7Bln%5E2%28x%29%7D%7Bx%7D++-+frac%7B2ln%28x%29%7D%7Bx%7D%2B+2intlimits+%7B+frac%7B1%7D%7Bx%7D+%2A+frac%7B1%7D%7Bx%7D+%7D+%2C+dx++%3D)
![= - frac{ln^2(x)}{x} - frac{2ln(x)}{x}+ 2intlimits {x^{-2} } , dx
= - frac{ln^2(x)}{x} - frac{2ln(x)}{x}+ 2*frac{x^{-2+1}}{-2+1}+C = = - frac{ln^2(x)}{x} - frac{2ln(x)}{x}+ 2intlimits {x^{-2} } , dx
= - frac{ln^2(x)}{x} - frac{2ln(x)}{x}+ 2*frac{x^{-2+1}}{-2+1}+C =](https://tex.z-dn.net/?f=%3D+-+frac%7Bln%5E2%28x%29%7D%7Bx%7D++-+frac%7B2ln%28x%29%7D%7Bx%7D%2B+2intlimits+%7Bx%5E%7B-2%7D+%7D+%2C+dx++%0A%3D+-+frac%7Bln%5E2%28x%29%7D%7Bx%7D++-+frac%7B2ln%28x%29%7D%7Bx%7D%2B+2%2Afrac%7Bx%5E%7B-2%2B1%7D%7D%7B-2%2B1%7D%2BC++%3D)
![= - frac{ln^2(x)}{x} - frac{2ln(x)}{x}-frac{2}{x}+C
=- frac{ln^2(x)+2ln(x)+2}{x}+C= = - frac{ln^2(x)}{x} - frac{2ln(x)}{x}-frac{2}{x}+C
=- frac{ln^2(x)+2ln(x)+2}{x}+C=](https://tex.z-dn.net/?f=%3D+-+frac%7Bln%5E2%28x%29%7D%7Bx%7D++-+frac%7B2ln%28x%29%7D%7Bx%7D-frac%7B2%7D%7Bx%7D%2BC%0A%3D-+frac%7Bln%5E2%28x%29%2B2ln%28x%29%2B2%7D%7Bx%7D%2BC%3D)
3)
![intlimits {x^2sin(2x)} , dx = intlimits {x^2} , d( -frac{cos(2x)}{2} ) =
-frac{x^2cos(2x)}{2}+ frac{1}{2} intlimits {cos(2x)} , d(x^2) = intlimits {x^2sin(2x)} , dx = intlimits {x^2} , d( -frac{cos(2x)}{2} ) =
-frac{x^2cos(2x)}{2}+ frac{1}{2} intlimits {cos(2x)} , d(x^2) =](https://tex.z-dn.net/?f=+intlimits+%7Bx%5E2sin%282x%29%7D+%2C+dx+%3D+intlimits+%7Bx%5E2%7D+%2C+d%28+-frac%7Bcos%282x%29%7D%7B2%7D+%29+%3D%0A+-frac%7Bx%5E2cos%282x%29%7D%7B2%7D%2B+frac%7B1%7D%7B2%7D+intlimits+%7Bcos%282x%29%7D+%2C+d%28x%5E2%29+++%3D)
![= -frac{x^2cos(2x)}{2}+ intlimits {xcos(2x)} , dx =
-frac{x^2cos(2x)}{2}+ frac{1}{2} intlimits {x} , d(sin(2x)) = = -frac{x^2cos(2x)}{2}+ intlimits {xcos(2x)} , dx =
-frac{x^2cos(2x)}{2}+ frac{1}{2} intlimits {x} , d(sin(2x)) =](https://tex.z-dn.net/?f=%3D+-frac%7Bx%5E2cos%282x%29%7D%7B2%7D%2B+intlimits+%7Bxcos%282x%29%7D+%2C+dx+++%3D%0A+-frac%7Bx%5E2cos%282x%29%7D%7B2%7D%2B++frac%7B1%7D%7B2%7D+intlimits+%7Bx%7D+%2C+d%28sin%282x%29%29+++%3D)
![= -frac{x^2cos(2x)}{2}+ frac{1}{2}[xsin(2x)- intlimits {sin(2x)} , dx] = = -frac{x^2cos(2x)}{2}+ frac{1}{2}[xsin(2x)- intlimits {sin(2x)} , dx] =](https://tex.z-dn.net/?f=%3D+-frac%7Bx%5E2cos%282x%29%7D%7B2%7D%2B++frac%7B1%7D%7B2%7D%5Bxsin%282x%29-+intlimits+%7Bsin%282x%29%7D+%2C+dx%5D+++%3D)
![= -frac{x^2cos(2x)}{2}+ frac{xsin(2x)}{2}- frac{1}{4}* intlimits {sin(2x)} , d(2x) = = -frac{x^2cos(2x)}{2}+ frac{xsin(2x)}{2}- frac{1}{4}* intlimits {sin(2x)} , d(2x) =](https://tex.z-dn.net/?f=%3D+-frac%7Bx%5E2cos%282x%29%7D%7B2%7D%2B++frac%7Bxsin%282x%29%7D%7B2%7D-+frac%7B1%7D%7B4%7D%2A++intlimits+%7Bsin%282x%29%7D+%2C+d%282x%29+++%3D)
![= -frac{x^2cos(2x)}{2}+ frac{xsin(2x)}{2}+ frac{1}{4}*cos(2x)+C= = -frac{x^2cos(2x)}{2}+ frac{xsin(2x)}{2}+ frac{1}{4}*cos(2x)+C=](https://tex.z-dn.net/?f=%3D+-frac%7Bx%5E2cos%282x%29%7D%7B2%7D%2B++frac%7Bxsin%282x%29%7D%7B2%7D%2B+frac%7B1%7D%7B4%7D%2Acos%282x%29%2BC%3D)
![= frac{1-2x^2}{4}*cos(2x)+ frac{x}{2}*sin(2x)+C = frac{1-2x^2}{4}*cos(2x)+ frac{x}{2}*sin(2x)+C](https://tex.z-dn.net/?f=%3D+frac%7B1-2x%5E2%7D%7B4%7D%2Acos%282x%29%2B++frac%7Bx%7D%7B2%7D%2Asin%282x%29%2BC)
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