Question

What is the integral of $\dfrac{1}{{{x^5}}}$?

Answer 1

Hint: The integration is nothing but the reverse process of differentiation. Here we are asked to find the integration of $\dfrac{1}{{{x^5}}}$. Also, to find the required answer, we need to apply the power rule. The power rule is the basic rule for integration. This power rule will increase the value of power and the coefficient is the power increased by one.
Formula:
The power rule of integration,
$\int {{x^n} = \dfrac{{{x^{n + 1}}}}{{n + 1}}} $, where $n$is the integer.

Complete step-by-step answer:
Given,
The term which is given to be evaluated is $\dfrac{1}{{{x^5}}}$.
Let the given term be assumed as $I$.
$I = \dfrac{1}{{{x^5}}}$
This is just a simple value that has only one term.
To find the integration always we need to add integration symbol and differentiation symbol, this we get,
$I = \int {\dfrac{1}{{{x^5}}}dx} $
To integrate the above term, we can use the power rule. The power rule is the basic rule for integration. This power rule will increase the value of power and the coefficient is the power increased by one.
The power rule of integration,
$\int {{x^n} = \dfrac{{{x^{n + 1}}}}{{n + 1}}} $, where $n$is the integer.
As the given term is in the denominator we need to make it to the numerator.
The positive power in the denominator if it goes to the numerator it will become negative.$I = \int {{x^{ - 5}}dx} $
As we compare the $n$values, we get $n = - 5$.
As we integrate the above equation with respect to the power rule, we get
\[I = \dfrac{{{x^{ - 5 + 1}}}}{{ - 5 + 1}}\]
As we add the power in the degrees we get
\[I = \dfrac{{{x^{ - 4}}}}{{ - 5 + 1}}\]
As we add the terms in the denominator, we get
\[I = \dfrac{{{x^{ - 4}}}}{{ - 4}}\]+C
The integration of the term $\dfrac{1}{{{x^5}}}$is \[ - \dfrac{{{x^{ - 4}}}}{4}\]+C

Note: To find the integration, we need to apply some rules of integration. Here we applied the power rule.
The power rule is compulsory because no integration will be done without this power rule. Also, we can use the power rule in differentiation. Therefore, the integration of the term $\dfrac{1}{{{x^5}}}$is \[ - \dfrac{{{x^{ - 4}}}}{4}\]