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.
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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.
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At 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.