Question

What is the antiderivative of \[\dfrac{1}{{{x}^{5}}}\]?

Answer 1

Hint: In order to find the antiderivative of \[\dfrac{1}{{{x}^{5}}}\], we will be applying the power rule to the function and upon solving it, we obtain the required anti derivative of \[\dfrac{1}{{{x}^{5}}}\]. We can also solve this function by applying the reverse process of differentiation.

Complete step-by-step answer:
Now, let us learn more about anti derivatives. An anti derivative is an inverse derivative, primitive function, primitive integral or indefinite integral of a function f is a differentiable function \[F\] whose derivative is equal to the original function \[f\]. This can be stated symbolically as \[F\grave{\ }=f\].
Now let us start finding the antiderivative of \[\dfrac{1}{{{x}^{5}}}\].
Now let us apply the reverse process of differentiation.
For example, we have \[{{x}^{a}}=\dfrac{{{x}^{a}}+1}{a+1}+C\]
We have this because, \[\dfrac{d}{dx}\left( \dfrac{1}{a+1}{{x}^{a+1}}+C \right)=\dfrac{a+1}{a+1}{{x}^{a+1-1}}={{x}^{a}}\]
Now we will be applying this formula to our given function i.e. \[\dfrac{1}{{{x}^{5}}}\].
We can write \[\dfrac{1}{{{x}^{5}}}\] as \[{{x}^{-5}}\]
Upon computing this according to the formula, we obtain
\[{{x}^{-5}}=\dfrac{1}{-5+1}{{x}^{-5+1}}+C\]
\[=\dfrac{1}{-4}{{x}^{-4}}+C\]
We can write this in the simple form in the following way,
\[-\dfrac{1}{4{{x}^{4}}}+C\].
\[\therefore \] The anti derivative of \[\dfrac{1}{{{x}^{5}}}\] is \[-\dfrac{1}{4{{x}^{4}}}+C\].

Note: An integral usually has a defined limit where as an anti derivative is usually a general case and will most always have a +C, the constant of integration, at the end of it. This is the only difference between the two other than that they are completely the same. One of the most common errors would be choosing of the proper integration method so it is very important to choose properly.