Section 5.2 : Computing Indefinite Integrals
9. Evaluate \( \displaystyle \int{{\frac{7}{{3{y^6}}} + \frac{1}{{{y^{10}}}} - \frac{2}{{\sqrt[3]{{{y^4}}}}}\,dy}}\).
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We first need to convert the root to a fractional exponent and move the \(y\)’s in the denominator to the numerator with negative exponents.
\[\int{{\frac{7}{{3{y^6}}} + \frac{1}{{{y^{10}}}} - \frac{2}{{\sqrt[3]{{{y^4}}}}}\,dy}} = \int{{\frac{7}{{3{y^6}}} + \frac{1}{{{y^{10}}}} - \frac{2}{{{y^{\frac{4}{3}}}}}\,dy}} = \int{{\frac{7}{3}{y^{ - 6}} + {y^{ - 10}} - 2{y^{ - \frac{4}{3}}}\,dy}}\]Remember that the “3” in the denominator of the first term stays in the denominator and does not move up with the \(y\).
Show Step 2Once we’ve gotten the root converted to a fractional exponent and the \(y\)’s out of the denominator there really isn’t too much to do other than to evaluate the integral.
\[\begin{align*}\int{{\frac{7}{{3{y^6}}} + \frac{1}{{{y^{10}}}} - \frac{2}{{\sqrt[3]{{{y^4}}}}}\,dy}} & = \int{{\frac{7}{3}{y^{ - 6}} + {y^{ - 10}} - 2{y^{ - \,\frac{4}{3}}}\,dy}}\\ & = \frac{7}{3}\left( {\frac{1}{{ - 5}}} \right){y^{ - 5}} + \left( {\frac{1}{{ - 9}}} \right){y^{ - 9}} - 2\left( { - \frac{3}{1}} \right){y^{ - \,\frac{1}{3}}} + c\\ & = \require{bbox} \bbox[2pt,border:1px solid black]{{ - \frac{7}{{15}}{y^{ - 5}} - \frac{1}{9}{y^{ - 9}} + 6{y^{ - \,\frac{1}{3}}} + c}}\end{align*}\]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.