Question

What is the antiderivative of $\dfrac{1}{x^{2}}$ ?

Answer 1

In this question, we need to find the antiderivative of $\dfrac{1}{x^{2}}$ . The anti-derivative of a function is nothing but its derivative is equal to its original function. Anti derivative is otherwise known as integrating the function . The inverse of differentiation is known as integral. The symbol `$\int$’ is the sign of integration. The process of finding the integral is known as integration. With the help of the reverse power rule, we can find the antiderivative of the given function.
Reverse power rule :
Reverse power rule is used to integrate the expression which is in the form of $x^{n}\$.
$\int x^{n}dx = \dfrac{x^{n + 1}}{n + 1} + c$
Where $c$ is the constant of integration.
This rule is not applicable when $n = - 1$ .

Complete step-by-step solution:
Given, $\dfrac{1}{x^{2}}$
Now we need to rewrite $\dfrac{1}{x^{2}}$ in the form of $x^{n}$ .
$\dfrac{1}{x^{2}} = x^{- 2}$
On integrating,
We get,
$I = \int x^{- 2}{dx}$
By using reverse power rule,
We get,
$I = \dfrac{x^{- 2 + 1}}{- 2 + 1} + c$
Where $c$ is the constant of integration.
On simplifying ,
We get,
$I = \dfrac{x^{- 1}}{- 1} + c$
Thus we get,
$I = - \dfrac{1}{x} + c$
Therefore the antiderivative of $\dfrac{1}{x^{2}}\$ is $- \dfrac{1}{x} + c$
Final answer :
The antiderivative of $\dfrac{1}{x^{2}}$ is $- \dfrac{1}{x} + c$

Note: Anti derivative is also known as inverse derivative. The concept used in this question is integration method, that is integration by using reverse power rule . Since this is an indefinite integral we have to add an arbitrary constant `$c$’. $c$ is called the constant of integration. The variable $x$ in ${dx}$ is known as the variable of integration or integrator.