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### 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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Hint : Don’t forget to convert the root to a fractional exponents and move the $$y$$’s in the denominator to the numerator with negative exponents.
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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 $$y$$.

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Once 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.