Integrale gebrochen rationale Funktionen
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Exercise:
prvmulticols nprvmulticols abcliste abc displaystyle fracx+x+ddx abc displaystyle fracx+x+ddx abc displaystyle fracfracx+fracfracx+fracddx abc displaystyle fracay+bcy+dddy abc displaystyle fracbs+tbu+bvddb abc displaystyle fracs^t+utr^+rddt abcliste multicols
Solution:
abcliste %a abc ed fracx+x+ddx !left-fracx+rightddx x-ln|x+|+C ed %b abc ed fracx+x+ frac+fractfracx+ Rightarrowquad fracx+x+ddx fracx+fracfracddxx+ fracx+fracln|x+|+C ed %c abc ed fracfracx+fracfracx+frac frac+frac-tfractfracx+tfrac Rightarrowquad fracfracx+fracfracx+fracddx fracx-fracfracddxtfracx+tfrac fracx-fracln!left|tfracx+tfracright|+C ed %d abc ed fracay+bcy+d fracac+fracbc-adcfraccy+d Rightarrowquad fracay+bcy+dddy fracacy+fracbc-adcfracddycy+d fracacy+fracbc-adc^ln|cy+d|+C ed %e abc ed fracbs+tbu+bv fracsu+v+fractu+vfracb Rightarrowquad fracbs+tbu+bvddb fracsu+vb+fractu+vln|b|+C ed %f abc ed fracs^ t+ur^ t+r fracs^r^+fracu-fracs^r^r^ t+r Rightarrowquad fracs^t+ur^t+rddt fracs^r^t+leftu-fracs^r^right!fracddtr^ t+r fracs^r^t+fracu-fracs^r^r^ln|r^ t+r|+C ed abcliste
prvmulticols nprvmulticols abcliste abc displaystyle fracx+x+ddx abc displaystyle fracx+x+ddx abc displaystyle fracfracx+fracfracx+fracddx abc displaystyle fracay+bcy+dddy abc displaystyle fracbs+tbu+bvddb abc displaystyle fracs^t+utr^+rddt abcliste multicols
Solution:
abcliste %a abc ed fracx+x+ddx !left-fracx+rightddx x-ln|x+|+C ed %b abc ed fracx+x+ frac+fractfracx+ Rightarrowquad fracx+x+ddx fracx+fracfracddxx+ fracx+fracln|x+|+C ed %c abc ed fracfracx+fracfracx+frac frac+frac-tfractfracx+tfrac Rightarrowquad fracfracx+fracfracx+fracddx fracx-fracfracddxtfracx+tfrac fracx-fracln!left|tfracx+tfracright|+C ed %d abc ed fracay+bcy+d fracac+fracbc-adcfraccy+d Rightarrowquad fracay+bcy+dddy fracacy+fracbc-adcfracddycy+d fracacy+fracbc-adc^ln|cy+d|+C ed %e abc ed fracbs+tbu+bv fracsu+v+fractu+vfracb Rightarrowquad fracbs+tbu+bvddb fracsu+vb+fractu+vln|b|+C ed %f abc ed fracs^ t+ur^ t+r fracs^r^+fracu-fracs^r^r^ t+r Rightarrowquad fracs^t+ur^t+rddt fracs^r^t+leftu-fracs^r^right!fracddtr^ t+r fracs^r^t+fracu-fracs^r^r^ln|r^ t+r|+C ed abcliste