# Integral of (x+1)/(x^2)

## \int\frac{x+1}{x^2}dx

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$-\frac{1}{x}+\ln\left|x\right|+C_0$

## Step by step solution

Problem

$\int\frac{x+1}{x^2}dx$
1

Split the fraction $\frac{x+1}{x^2}$ in two terms with same denominator

$\int\left(\frac{1}{x^2}+\frac{x}{x^2}\right)dx$
2

Simplifying the fraction by $x$

$\int\left(\frac{1}{x^2}+\frac{1}{x}\right)dx$
3

The integral of a sum of two or more functions is equal to the sum of their integrals

$\int\frac{1}{x^2}dx+\int\frac{1}{x}dx$
4

The integral of the inverse of the lineal function is given by the following formula, $\displaystyle\int\frac{1}{x}dx=\ln(x)$

$\int\frac{1}{x^2}dx+\ln\left|x\right|$
5

Rewrite the exponent using the power rule $\frac{a^m}{a^n}=a^{m-n}$, where in this case $m=0$

$\int x^{-2}dx+\ln\left|x\right|$
6

Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a constant function

$\frac{x^{-1}}{-1}+\ln\left|x\right|$
7

Apply the formula: $\frac{x}{-1}$$=-x$, where $x=x^{-1}$

$\ln\left|x\right|-x^{-1}$
8

Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number

$\ln\left|x\right|-\frac{1}{x}$
9

$-\frac{1}{x}+\ln\left|x\right|+C_0$

$-\frac{1}{x}+\ln\left|x\right|+C_0$

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### Main topic:

Integral calculus

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