Examples of Power Rule of Integration

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

Consider the function to be integrate

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 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 integrate

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}

Comments

comments