Find the integral

Factor the trinomial $x^2+x-2$ finding two numbers that multiply to form $-2$ and added form $1$

Rewrite the polynomial as the product of two binomials consisting of the sum of the variable and the found values

Simplifying

Expand the fraction $\frac{x-1}{x+3}$ into $2$ simpler fractions with common denominator $x+3$

Expand the integral $\int\left(\frac{x}{x+3}+\frac{-1}{x+3}\right)dx$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately

We can solve the integral $\int\frac{x}{x+3}dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $x+3$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part

Gather the results of all integrals

We can solve the integral $\int\frac{-1}{x+3}dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $x+3$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part

Substituting $u$ and $dx$ in the integral and simplify

Gather the results of all integrals

Combining like terms $-3\ln\left(x+3\right)$ and $-\ln\left(x+3\right)$

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

We can combine and rename $3$ and $C_0$ as other constant of integration