integral of ((x-1))/(x^2) — step-by-step solution

Let's integrate:

\int \frac{x - 1}{x^{2}} d x

For

x

≠ 0, divide each numerator term by the power in the denominator:

\frac{x - 1}{x^{2}} = \frac{1}{x} - \frac{1}{x^{2}}

ⅰ

\int \frac{x - 1}{x^{2}} d x

\int \frac{1}{x} d x - \int \frac{1}{x^{2}} d x

\int \frac{1}{x} d x - \int \frac{1}{x^{2}} d x

ⓘ The derivative of

\ln | x |

\frac{1}{x}

, so

\ln | x |

is an antiderivative of

\frac{1}{x}

\int \frac{1}{x} d x

\ln | x |

■An antiderivative of x^n is x^(n+1)/(n+1), for n ≠ -1:

- \int \frac{1}{x^{2}} d x

- \int x^{-2} d x

x^{-1}

\frac{1}{x}

\ln | x | + \frac{1}{x}

▶Indefinite integrals are defined up to an additive constant C, so this yields:

\frac{1}{x} + \ln | x | + C