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Section 5.2 : Computing Indefinite Integrals
11. Evaluate \( \displaystyle \int{{\sqrt z \left( {{z^2} - \frac{1}{{4z}}} \right)\,dz}}\).
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Hint : Remember that there is no “Product Rule” for integrals and so we’ll need to eliminate the product before integrating.
Since there is no “Product Rule” for integrals we’ll need to multiply the terms out prior to integration.
\[\int{{\sqrt z \left( {{z^2} - \frac{1}{{4z}}} \right)\,dz}} = \int{{{z^{^{\frac{5}{2}}}} - \frac{1}{{4{z^{^{\frac{1}{2}}}}}}\,dz}} = \int{{{z^{^{\frac{5}{2}}}} - \frac{1}{4}{z^{^{ - \,\frac{1}{2}}}}\,dz}}\]Don’t forget to convert the root to a fractional exponent and move the \(z\)’s out of the denominator.
Show Step 2At this point there really isn’t too much to do other than to evaluate the integral.
\[\int{{\sqrt z \left( {{z^2} - \frac{1}{{4z}}} \right)\,dz}} = \int{{{z^{^{\frac{5}{2}}}} - \frac{1}{4}{z^{^{ - \,\frac{1}{2}}}}\,dz}} = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{2}{7}{z^{^{\frac{7}{2}}}} - \frac{1}{2}{z^{^{\frac{1}{2}}}} + c}}\]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.