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Verifica primitiva per derivazione 1

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Exercise

https://texercises.raemilab.ch/exercise/verifica-primitiva-per-derivazione-1/

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Exercise:
Determina la primitiva di ln^x sapo che è della forma x aln^x+blnx+c con abcinmathbbR. Poi sfrutta un ragionamento analogo per determinare la primitiva di ln^x.

Solution:
Derivando l'espressione data otteniamo * x aln^x+blnx+c'aln^x+blnx+c+x leftafraclnxx+fracbxright aln^x+a+blnx+b+c * Confrontando i coefficeitni otteniamo a b- e c dunque la primitiva è xleftln^x-lnx+right+C. Generalizziamo subito al grado derivando * x aln^x+bln^x+clnx+d'aln^x+bln^x+clnx+d &qquad +x leftafracln^xx+bfraclnxx+fraccxright aln^x+a+bln^x+b+clnx+c+d * da cui otteniamo a b- c e d-. textbfIn generale: possiamo calcolare la primitiva di ln^nx pono come Ansatz xa_nln^nx+a_n-ln^n-x+s +a_lnx+a_x_k^na_kln^kx e calcolando la derivata: * leftx_k^na_kln^kxright'_k^na_kln^kx+_k^n a_k kln^k-x_k^n-a_k+k+a_k+ln^kx+a_nln^nx * Così che a_n e a_k-k+a_k+-^n-kfracn!k! e quindi la primitiva è x_k^n-^n-kfracn!k!ln^kx.

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Tags	Calcolo integrale, Verifica primitiva per derivazione
Difficulty	(2, default)
Points	0 (default)
Language	(Italiano)
Type	Calculative / Quantity
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