# Integral of 1/x^n (1 by x power n): Formula, Proof

The integral of 1/x^n (1 by x power n) is equal to x-n+1/(1-n) +C. Here we will learn how to integrate 1/xn using the power rule of integration.

The integration formula of 1/xn is given by

$\int \dfrac{dx}{x^n}=\dfrac{x^{1-n}}{1-n}+C$

where C denotes an integral constant.

## Integration of 1/xn

The integral of 1/xn is denoted by ∫dx/xn. We have:

$\int \dfrac{dx}{x^n}$

= $\int x^{-n} \, dx$ by the rule of indices.

= $\dfrac{x^{-n+1}}{1-n}+C$ by the power rule of integrations.

So the integration of 1 by x^n is equal to x1-n/(1-n) +C (C is a constant), and this is obtained by the power rule of integrals.

## Definite Integral of 1/xn

Question: Find the integration of 1/xn from 0 to 1, that is,

Find $\int_0^1 \dfrac{dx}{x^n}$.

AS the integration of 1/xn is equal to x-n+1/(1-n) +C by above, we obtain that

$\int_0^1 \dfrac{dx}{x^n}$

= $\Big[\dfrac{x^{1-n}}{1-n} \Big]_0^1$

= $\dfrac{1^{1-n}}{1-n} – \dfrac{0^{1-n}}{1-n}$

= $\dfrac{1}{1-n} – 0$

= $\dfrac{1}{1-n}$.

So the integration of 1/xn from 0 to 1 is equal to 1/(n-1).

More Reading: How to integrate 2x

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Integration of square root of a2-x2

## FAQs

### Q1: What is the integration of 1/xn?

Answer: The integration of 1/xn is equal to x1-n/(1-n) +C where C is an arbitrary integral constant.