$\int (\frac{7}{x^6}-\frac{5}{x^4}+\frac{3}{x^2}-70x)dx=$

$\int (\frac{7}{x^6}-\frac{5}{x^4}+\frac{3}{x^2}-70x)dx$

$=\int 7x^{-6}dx-\int 5x^{-4}dx+\int 3x^{-2}dx-\int 70xdx$

$=\int x^{-6}dx-5\int x^{-4}dx+3\int x^{-2}dx-70\int xdx$

$=7\left(\frac{x^{-6+1}}{-6+1}\right)-5\left(\frac{x^{-4+1}}{-4+1}\right)+3\left(\frac{x^{-2+1}}{-2+1}\right)-70\left(\frac{x^{1+1}}{1+1}\right)+c$

$=7\left(\frac{x^{-5}}{-5}\right)-5\left(\frac{x^{-3}}{-3}\right)+3\left(\frac{x^{-1}}{-1}\right)-70\left(\frac{x^2}{2}\right)+c$

$=\frac{-7}{5x^5}+\frac{5}{3x^3}-\frac{3}{x}-35x^2+c$