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Unfortunately, this means that the site will be down during this time. I apologize for any inconvenience this might cause.
Paul
February 18, 2026
Section 5.2 : Computing Indefinite Integrals
12. Evaluate \( \displaystyle \int{{\frac{{{z^8} - 6{z^5} + 4{z^3} - 2}}{{{z^4}}}\,dz}}\).
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Since there is no “Quotient Rule” for integrals we’ll need to break up the integrand and simplify a little prior to integration.
\[\int{{\frac{{{z^8} - 6{z^5} + 4{z^3} - 2}}{{{z^4}}}\,dz}} = \int{{\frac{{{z^8}}}{{{z^4}}} - \frac{{6{z^5}}}{{{z^4}}} + \frac{{4{z^3}}}{{{z^4}}} - \frac{2}{{{z^4}}}\,dz}} = \int{{{z^4} - 6z + \frac{4}{z} - 2{z^{ - 4}}\,dz}}\] Show Step 2At this point there really isn’t too much to do other than to evaluate the integral.
\[\int{{\frac{{{z^8} - 6{z^5} + 4{z^3} - 2}}{{{z^4}}}\,dz}} = \int{{{z^4} - 6z + \frac{4}{z} - 2{z^{ - 4}}\,dz}} = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{1}{5}{z^5} - 3{z^2} + 4\ln \left| z \right| + \frac{2}{3}{z^{ - 3}} + c}}\]Don’t forget to add on the “+c” since we know that we are asking what function did we differentiate to get the integrand and the derivative of a constant is zero and so we do need to add that onto the answer.