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}