Question

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

Answer 1

Antiderivative of a function.

The antiderivative of a variable $x$ raised power of $n$ is computed by using the power rule.

According to the power rule,

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

We have to find the antiderivative of $\frac{1}{x^2}$.

$$\begin{gathered} \therefore \int \frac{1}{x^2}dx=\int x^{-2}dx\left[\because \frac{1}{x^n}=x^{-n}\right] \\ \Rightarrow =\frac{x^{-2+1}}{-2+1}\left[n=-2\right] \\ \Rightarrow =\frac{x^{-1}}{-1} \\ \Rightarrow =-\frac{1}{x}+C \end{gathered}$$

Hence, $\int \frac{1}{x^2}dx=-\frac{1}{x}+C$ , Where $C$ is a constant.