Question

determine a integral de ∫(3x^-4 - 3x+4)dx
por favor me expliquem o processo

Answer 1

Calcular a integral indefinida:

     $\displaystyle\int (3x^{-4}-3x+4)\,dx\\\\\\ =3\int x^{-4}\,dx-3\int x\,dx+4\int dx$


Aqui, basta você usar a regra para encontrar primitivas de potências:

     $\displaystyle\int x^n\,dx=\left\{\!\begin{array}{ll}\dfrac{x^{n+1}}{n+1}+C\,,&\quad\textsf{se~~}n\ne -1\\\\ \mathrm{\ell n}|x|+C\,,&\quad\textsf{se~~}n=-1 \end{array}\right.$


Então, a integral fica

     $=3\cdot \dfrac{x^{-4+1}}{-4+1}-3\cdot \dfrac{x^{1+1}}{1+1}+4x+C\\\\\\ =3\cdot \dfrac{x^{-3}}{-3}-3\cdot \dfrac{x^2}{2}+4x+C\\\\\\ =-x^{-3}-\dfrac{3}{2}\,x^2+4x+C$

     $=-\,\dfrac{1}{x^3}-\dfrac{3}{2}\,x^2+4x+C\quad\longleftarrow\quad\textsf{resposta.}$


Bons estudos! :-)