Question

How do you integrate\[\int{\dfrac{1}{{{x}^{4}}}dx}\]?

Answer 1

Hint: In the given question, we have been asked to integrate the following function. In order to solve the question, we need to use the formulas and the concepts of integration. First we need to convert the fraction in integral by using the formula of integration I.e. \[\dfrac{1}{{{x}^{a}}}={{x}^{-a}}\] and then by applying \[\int{{{x}^{a}}dx}=\dfrac{{{x}^{a+1}}}{a+1}+C\], we will get our required answer.

Complete step by step solution:
We have given,
\[\Rightarrow \int{\dfrac{1}{{{x}^{4}}}dx}\]
As we know that,
Using the formulas of integration, i.e.
\[\dfrac{1}{{{x}^{a}}}={{x}^{-a}}\]
Applying this formula of integration in the given expression, we get
\[\Rightarrow \int{\dfrac{1}{{{x}^{4}}}dx}=\int{{{x}^{-4}}dx}\]
Now, using the integration formulas, i.e.
\[\int{{{x}^{a}}dx}=\dfrac{{{x}^{a+1}}}{a+1}+C\], where ‘a’ should not be equal to -1 and C is the constant of the given integration.
Applying this in the above given integral, we get
\[\Rightarrow \int{\dfrac{1}{{{x}^{4}}}dx}=\int{{{x}^{-4}}dx}\]
\[\Rightarrow \int{{{x}^{-4}}dx}=\dfrac{{{x}^{-4+1}}}{-4+1}+C\]
Simplifying the above integral, we get
\[\Rightarrow \dfrac{{{x}^{-4+1}}}{-4+1}+C=\dfrac{{{x}^{-3}}}{-3}+C=-\dfrac{1}{3}{{x}^{-3}}+C\]
Simplifying the above expression, we get
\[\Rightarrow -\dfrac{1}{3{{x}^{3}}}+C\]

Therefore,
\[\Rightarrow \int{\dfrac{1}{{{x}^{4}}}dx}=-\dfrac{1}{3{{x}^{3}}}+C\]
Hence, it is the required integration.

Note: Integration is defined as the summation of all the discrete data. In order to solve the question we should remember the basic property or the formulas of integration, this would make it easier to solve the question. We should do all the calculations carefully and explicitly to avoid making errors. While solving these types of questions, first we always need to take out the constant part of the integration and then we will integrate the variable part using a suitable integration formula.