Integrale mit Polynomdivision ohne Rest
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Exercise:
nprvmulticols abcliste %a abc displaystyle frac x^ + x y^ + x y + x + y^ + yy^+x+ddy %b abc displaystyle frac y^ + y^ + y + y^ + y + ddy %c abc displaystyle frac t^ + t^ + t^ - t + t^ + t + ddt %d abc displaystyle frac s^ + s^ - s^ - s + s^ + s^ - s + dds %e abc displaystyle frac r^ - r^ + r^ - r + r^ - r + ddr %f abc displaystyle frac u^ + u^ + u^ + u + u^ + u^ + u + ddu %g abc displaystyle fracd^ x^ + e d x^ - d x + e d x + e^ x^ - edx+eddx %h abc displaystyle fraca c x^ + a d x^ + a e x + b c x^ + b d x + b ea x + bddx %i abc displaystyle fraca c y^ + a d y^ + a e y + b c y^ + b d y + b ec y^ + d y + eddy %j abc displaystyle fraca c t^ + a d t^ + b c t^ + b da t^ + bddt %k abc displaystyle fraca d r^ + a e r^ + b d r^ + b e r^ + c d r + c ed r + eddr %l abc displaystyle fraca d s^ + a e s + b d s + b ea s + bdds abcliste nprvmulticols
Solution:
abcliste % a abc ed frac x^ + x y^ + x y + x + y^ + yy^+x+ddy bigly + xbigrddy tfrac y^ + xy + C ed % b abc ed frac y^ + y^ + y + y^ + y + ddy bigly + bigrddy tfrac y^ + y + C ed % c abc ed frac t^ + t^ + t^ - t + t^ + t + ddt biglt^ - t + bigrddt tfract^ - tfrac t^ + t + C ed % d abc ed frac s^ + s^ - s^ - s + s^ + s^ - s + dds bigls + bigrdds tfrac s^ + s + C ed % e abc ed frac r^ - r^ + r^ - r + r^ - r + ddr biglr^ + r + bigrddr tfracr^ + tfrac r^ + r + C ed % f abc ed frac u^ + u^ + u^ + u + u^ + u^ + u + ddu biglu + bigrddu tfrac u^ + u + C ed % g abc ed fracd^ x^ + e d x^ - d x + e d x + e^ x^ - edx+eddx bigle x^ + d x - bigrddx tfracex^ + tfracdx^ - x + C ed % h abc ed fraca c x^ + a d x^ + a e x + b c x^ + b d x + b ea x + bddx biglc x^ + d x + ebigrddx tfraccx^ + tfracdx^ + e x + C ed % i abc ed fraca c y^ + a d y^ + a e y + b c y^ + b d y + b ec y^ + d y + eddy bigla y + bbigrddy tfracay^ + b y + C ed % j abc ed fraca c t^ + a d t^ + b c t^ + b da t^ + bddt biglc t^ + dbigrddt tfracct^ + d t + C ed % k abc ed fraca d r^ + a e r^ + b d r^ + b e r^ + c d r + c ed r + eddr bigla r^ + b r^ + cbigrddr tfracar^ + tfracbr^ + c r + C ed % l abc ed fraca d s^ + a e s + b d s + b ea s + bdds bigld s + ebigrdds tfracds^ + e s + C ed abcliste
nprvmulticols abcliste %a abc displaystyle frac x^ + x y^ + x y + x + y^ + yy^+x+ddy %b abc displaystyle frac y^ + y^ + y + y^ + y + ddy %c abc displaystyle frac t^ + t^ + t^ - t + t^ + t + ddt %d abc displaystyle frac s^ + s^ - s^ - s + s^ + s^ - s + dds %e abc displaystyle frac r^ - r^ + r^ - r + r^ - r + ddr %f abc displaystyle frac u^ + u^ + u^ + u + u^ + u^ + u + ddu %g abc displaystyle fracd^ x^ + e d x^ - d x + e d x + e^ x^ - edx+eddx %h abc displaystyle fraca c x^ + a d x^ + a e x + b c x^ + b d x + b ea x + bddx %i abc displaystyle fraca c y^ + a d y^ + a e y + b c y^ + b d y + b ec y^ + d y + eddy %j abc displaystyle fraca c t^ + a d t^ + b c t^ + b da t^ + bddt %k abc displaystyle fraca d r^ + a e r^ + b d r^ + b e r^ + c d r + c ed r + eddr %l abc displaystyle fraca d s^ + a e s + b d s + b ea s + bdds abcliste nprvmulticols
Solution:
abcliste % a abc ed frac x^ + x y^ + x y + x + y^ + yy^+x+ddy bigly + xbigrddy tfrac y^ + xy + C ed % b abc ed frac y^ + y^ + y + y^ + y + ddy bigly + bigrddy tfrac y^ + y + C ed % c abc ed frac t^ + t^ + t^ - t + t^ + t + ddt biglt^ - t + bigrddt tfract^ - tfrac t^ + t + C ed % d abc ed frac s^ + s^ - s^ - s + s^ + s^ - s + dds bigls + bigrdds tfrac s^ + s + C ed % e abc ed frac r^ - r^ + r^ - r + r^ - r + ddr biglr^ + r + bigrddr tfracr^ + tfrac r^ + r + C ed % f abc ed frac u^ + u^ + u^ + u + u^ + u^ + u + ddu biglu + bigrddu tfrac u^ + u + C ed % g abc ed fracd^ x^ + e d x^ - d x + e d x + e^ x^ - edx+eddx bigle x^ + d x - bigrddx tfracex^ + tfracdx^ - x + C ed % h abc ed fraca c x^ + a d x^ + a e x + b c x^ + b d x + b ea x + bddx biglc x^ + d x + ebigrddx tfraccx^ + tfracdx^ + e x + C ed % i abc ed fraca c y^ + a d y^ + a e y + b c y^ + b d y + b ec y^ + d y + eddy bigla y + bbigrddy tfracay^ + b y + C ed % j abc ed fraca c t^ + a d t^ + b c t^ + b da t^ + bddt biglc t^ + dbigrddt tfracct^ + d t + C ed % k abc ed fraca d r^ + a e r^ + b d r^ + b e r^ + c d r + c ed r + eddr bigla r^ + b r^ + cbigrddr tfracar^ + tfracbr^ + c r + C ed % l abc ed fraca d s^ + a e s + b d s + b ea s + bdds bigld s + ebigrdds tfracds^ + e s + C ed abcliste