The Power Rule of Integration

The power rule of integration is an important and fundamental formula in integral calculus. We already know that the 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, the integral sign $$\int {} $$ and $$\frac{d}{{dx}}$$ on the right side will cancel each other out, 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 separating the integral in 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 the power rule of integration, we have
\[\begin{gathered} I = 4\frac{{{x^{2 + 1}}}}{{2 + 1}} + 2\frac{{{x^{1 + 1}}}}{{1 + 1}} + c \\ \Rightarrow I = \frac{4}{3}{x^3} + {x^2} + c \\ \Rightarrow \int {\left( {4{x^2} + 2x} \right)dx} = \frac{4}{3}{x^3} + {x^2} + c \\ \end{gathered} \]