Answer
$3\displaystyle \ln|x|+\frac{1}{4x^{4}}-\frac{1}{6x^{6}}+C$
Work Step by Step
$\displaystyle \int(\frac{3}{x}-\frac{1}{x^{5}}+\frac{1}{x^{7}})dx =\qquad $
The integral of a sum is the sum of the integrals:
$=\displaystyle \int\frac{3}{x}dx-\int\frac{1}{x^{5}}dx+\int\frac{1}{x^{7}}dx$
We have an integral of a constant times a function...
Also, use: $\displaystyle \frac{1}{x^{n}}=x^{-n}$
$=3\displaystyle \int x^{-1}dx-\int x^{-5}dx+\int x^{-7}dx$
for $n=-1, \displaystyle \qquad \int x^{n}dx=\ln|x|+C$
for $n\neq-1,\quad \displaystyle \int x^{n}dx=\frac{x^{n+1}}{n+1}+C$
$=3\displaystyle \ln|x|-\frac{x^{-4}}{-4}+\frac{x^{-6}}{-6}+C$
$=3\displaystyle \ln|x|+\frac{1}{4x^{4}}-\frac{1}{6x^{6}}+C$
