Question

Fie f:(0, infinit )→R f (x)=x^3*lnx. Sa se determine primitiva F a lui f care verifica F (1)=0

Answer 1

$F(x)=\int x^3\cdot lnx\ dx= \int\left(\dfrac{x^4}4\right)'\cdot lnx\ dx=\dfrac{x^4}4\cdot lnx-\int\dfrac{x^4}4\cdot\dfrac{1}x\ dx=\\\\=\dfrac{x^4}4\cdot lnx-\dfrac{x^4}{16}+C.\ Pe\ C\ il\ afl\breve{a}m\ din\ condi\c{t}ia\ F(1)=0:\\\\F(1)=-\dfrac1{16},\ deci\ C=-\dfrac{1}{16}.\\\\F(x)=\dfrac{x^4}4\cdot lnx-\dfrac{x^4}{16}-\dfrac1{16}.$

Pentru aflarea primitivei am aplicat metoda integrării prin părți.

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