Power Rule of Integration

Power Rule of integration is an important and fundamental formula in integral calculus study. As we know that inverse process of differentiation is called integration.

The basic power rule of integration is of the form

\int  {{x^n}dx = } \frac{{{x^{n + 1}}}}{{n + 1}} + c,\,\,\,\,n \ne - 1

Now consider

\begin{gathered} \frac{d}{{dx}}\left[ {\frac{{{x^{n + 1}}}}{{n  + 1}} + c} \right] = \frac{d}{{dx}}\left[ {\frac{{{x^{n + 1}}}}{{n + 1}}}  \right] + \frac{d}{{dx}}\left( c \right) \\ \Rightarrow \frac{d}{{dx}}\left[  {\frac{{{x^{n + 1}}}}{{n + 1}} + c} \right] = n + 1\frac{{{x^{n + 1 - 1}}}}{{n  + 1}} + 0 \\ \Rightarrow \frac{d}{{dx}}\left[  {\frac{{{x^{n + 1}}}}{{n + 1}} + c} \right] = {x^n} \\ \Rightarrow {x^n} = \frac{d}{{dx}}\left[  {\frac{{{x^{n + 1}}}}{{n + 1}} + c} \right]\,\,\,\,\,\,\,{\text{ - - -  }}\left( {\text{i}} \right) \\ \end{gathered}


Integrating both sides of equation (i) with respect to x, we have

  \Rightarrow \int {{x^n}} dx = \int {\frac{d}{{dx}}\left[ {\frac{{{x^{n +  1}}}}{{n + 1}} + c} \right]}


Since integration and differentiation are reverse processes to each other , so the integral sign \int  {} and \frac{d}{{dx}} on the right side will cancel each other, i.e.

\int  {{x^n}} dx = \frac{{{x^{n + 1}}}}{{n + 1}} + c

Example: Evaluate the integral \int {\left( {4{x^2} +  2x} \right)dx} with respect to x
We have integral

I =  \int {\left( {4{x^2} + 2x} \right)dx}


In the given function we have two terms, now separate the integral on these two functions, we get

\begin{gathered} I = \int {4{x^2}dx + \int {2xdx} } \\ \Rightarrow I = 4\int {{x^2}dx + 2\int {xdx}  } \\ \end{gathered}

Now using power rule of integration, we have

\begin{gathered} I = 4\frac{{{x^{2 + 1}}}}{{2 + 1}} +  2\frac{{{x^{1 + 1}}}}{{1 + 1}} + c \\ \Rightarrow I = \frac{4}{3}{x^3} + {x^2} + c \\ \Rightarrow \int {\left( {4{x^2} + 2x}  \right)dx} = \frac{4}{3}{x^3} + {x^2} +  c \\ \end{gathered}

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