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Power rule of Integration

Formula

$\displaystyle \int{x^n\,}dx$ $\,=\,$ $\dfrac{x^{n+1}}{n+1}+c$

Introduction

When $x$ is used to represent a variable and $n$ represents a constant, an algebraic expression $x^n$ is formed in exponential form.

The algebraic expression is defined in variable $x$. So, the indefinite integration should be done with represent to $x$ and it is written in mathematical form as follows.

$\displaystyle \int{x^n\,}dx$

The indefinite integration of the function $x^n$ with respect to $x$ is equal to the sum of the quotient of $x$ raised to the power of $n+1$ by $n+1$ and the constant of integration, which is denoted by $c$ in mathematics.

$\implies$ $\displaystyle \int{x^n\,}dx$ $\,=\,$ $\dfrac{x^{n+1}}{n+1}+c$

It is called the power rule of integration. It is also called as the reverse power rule in calculus.

If $n = -1$, the right hand side expression of the equation become undefined. Hence, the value of exponent $n$ should not be equal to $-1$ ($n \ne -1$).

Alternative forms

The power rule of integration can be written in terms of any variable as exampled here.

$(1)\,\,\,$ $\displaystyle \int{l^k\,}dl$ $\,=\,$ $\dfrac{l^{k+1}}{k+1}+c$

$(2)\,\,\,$ $\displaystyle \int{r^i\,}dr$ $\,=\,$ $\dfrac{r^{i+1}}{i+1}+c$

$(3)\,\,\,$ $\displaystyle \int{y^m\,}dy$ $\,=\,$ $\dfrac{y^{m+1}}{m+1}+c$

Proof

Learn how to prove the power rule of the integration in integral calculus.