Recognize and Rewrite: We have the integral: int(1)/(x^(2))dx To solve this, we recognize that 1/x^2 is the same as x^(-2). So, we rewrite the integral as: int(x^(-2))dxApply Power Rule: Now, we apply the power rule for integration, which states that the integral of x^n with respect to x is x^(n+1)/(n+1) + C, where n ≠ -1. In our case, n = -2, so we add 1 to the exponent and divide by the new exponent: int(x^(-2))dx = x^(-2+1)/(-2+1) + CSimplify Expression: Simplifying the expression, we get: x^(-1)/(-1) + C Which is the same as: -1/x + C

Calculus

Find indefinite integrals using the substitution

Full solution

\int \frac{1}{x^{2}} \, dx

Recognize and Rewrite: We have the integral: $\newline$ $\int \frac{1}{x^{2}} \, dx$ $\newline$ To solve this, we recognize that $\frac{1}{x^2}$ is the same as $x^{-2}$ . $\newline$ So, we rewrite the integral as: $\newline$ $\int x^{-2} \, dx$
Apply Power Rule: Now, we apply the power rule for integration, which states that the integral of $x^n$ with respect to $x$ is $\frac{x^{n+1}}{n+1} + C$ , where $n \neq -1$ . In our case, $n = -2$ , so we add $1$ to the exponent and divide by the new exponent: $\int x^{-2}dx = \frac{x^{-2+1}}{-2+1} + C$
Simplify Expression: Simplifying the expression, we get: $\newline$ $x^{-1}/(-1) + C$ $\newline$ Which is the same as: $\newline$ $-1/x + C$