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### Section 5.2 : Computing Indefinite Integrals

7. Evaluate $$\displaystyle \int{{\sqrt {{x^7}} - 7\,\sqrt[6]{{{x^5}}} + 17\,\,\sqrt[3]{{{x^{10}}}}\,dx}}$$.

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Hint : Don’t forget to convert the roots to fractional exponents.
Start Solution

We first need to convert the roots to fractional exponents.

$\int{{\sqrt {{x^7}} - 7\,\sqrt[6]{{{x^5}}} + 17\,\,\sqrt[3]{{{x^{10}}}}\,dx}} = \int{{{x^{\frac{7}{2}}} - 7{{\left( {{x^5}} \right)}^{\frac{1}{6}}} + 17{{\left( {{x^{10}}} \right)}^{\frac{1}{3}}}\,dx}} = \int{{{x^{\frac{7}{2}}} - 7{x^{\frac{5}{6}}} + 17{x^{\frac{{10}}{3}}}\,dx}}$ Show Step 2

Once we’ve gotten the roots converted to fractional exponents there really isn’t too much to do other than to evaluate the integral.

\begin{align*}\int{{\sqrt {{x^7}} - 7\,\sqrt[6]{{{x^5}}} + 17\,\,\sqrt[3]{{{x^{10}}}}\,dx}} & = \int{{{x^{\frac{7}{2}}} - 7{x^{\frac{5}{6}}} + 17{x^{\frac{{10}}{3}}}\,dx}}\\ & = \frac{2}{9}{x^{\frac{9}{2}}} - 7\left( {\frac{6}{{11}}} \right){x^{\frac{{11}}{6}}} + 17\left( {\frac{3}{{13}}} \right){x^{\frac{{13}}{3}}} + c = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{2}{9}{x^{\frac{9}{2}}} - \frac{{42}}{{11}}{x^{\frac{{11}}{6}}} + \frac{{51}}{{13}}{x^{\frac{{13}}{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.