# Examples of the Power Rule of Integration

Example: Integrate $\frac{{{x^2} – 2{x^4}}}{{{x^4}}}$ with respect to $x$.

Consider the function to be integrated $I = \int {\frac{{{x^2} – 2{x^4}}}{{{x^4}}}dx}$
$\begin{gathered} I = \int {\left( {\frac{{{x^2}}}{{{x^4}}} – 2\frac{{{x^4}}}{{{x^4}}}} \right)dx} \\ \Rightarrow I = \int {\left( {\frac{1}{{{x^2}}} – 2} \right)dx} \\ \Rightarrow I = \int {{x^{ – 2}}dx – 2\int {dx} } \\ \end{gathered}$

Using the power rule of integration, we have
$\begin{gathered} I = \frac{{{x^{ – 2 + 1}}}}{{ – 2 + 1}} – 2x + c \\ \Rightarrow I = \frac{{{x^{ – 1}}}}{{ – 1}} – 2x + c \\ \Rightarrow I = – \frac{1}{x} – 2x + c \\ \end{gathered}$

Example: Integrate $\left( {\sqrt[3]{x} + \frac{1}{{\sqrt[3]{x}}}} \right)$ with respect to $x$.

Consider the function to be integrated $I = \int {\left( {\sqrt[3]{x} + \frac{1}{{\sqrt[3]{x}}}} \right)dx}$
$\begin{gathered} I = \int {\left( {{x^{\frac{1}{3}}} + \frac{1}{{{x^{\frac{1}{3}}}}}} \right)dx} \\ \Rightarrow I = \int {\left[ {{x^{\frac{1}{3}}} + {x^{ – \frac{1}{3}}}} \right]dx} \\ \Rightarrow I = \int {{x^{\frac{1}{3}}}dx + \int {{x^{ – \frac{1}{3}}}dx} } \\ \Rightarrow I = \frac{{{x^{\frac{1}{3} + 1}}}}{{\frac{1}{3} + 1}} + \frac{{{x^{ – \frac{1}{3} + 1}}}}{{ – \frac{1}{3} + 1}} + c \\ \Rightarrow I = \frac{{{x^{\frac{4}{3}}}}}{{\frac{4}{3}}} + \frac{{{x^{\frac{2}{3}}}}}{{\frac{2}{3}}} + c \\ \Rightarrow I = \frac{3}{4}{x^{\frac{4}{3}}} + \frac{3}{2}{x^{\frac{2}{3}}} + c \\ \end{gathered}$