Math Doubts

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.

Math Doubts
Math Doubts is a best place to learn mathematics and from basics to advanced scientific level for students, teachers and researchers. Know more
Follow us on Social Media
Math Problems

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.

Learn more