Recognize Simplification Opportunity: Recognize that the integral can be simplified by dividing the numerator by the denominator. I = ∫((x-3)/(x+3))dx We can perform long division or notice that (x-3)/(x+3) can be rewritten as 1 - 6/(x+3).Rewrite in Simplified Form: Rewrite the integral in terms of the simplified expression. I = ∫(1 - 6/(x+3))dx This separates the integral into two simpler integrals.Integrate First Term: Integrate the first term, which is the integral of 1 with respect to x. I = ∫1dx - ∫(6/(x+3))dx The integral of 1 with respect to x is x.Integrate Second Term: Integrate the second term, which is the integral of 6/(x+3) with respect to x. I = x - 6∫(1/(x+3))dx The integral of 1/(x+3) with respect to x is ln|x+3|.Combine and Add Constant: Combine the results of the integrals and include the constant of integration. I = x - 6ln|x+3| + C This is the indefinite integral of the given function.

Calculus

Find indefinite integrals using the substitution and by parts

Full solution

\int \frac{(x-3)\,dx}{x+3}

Recognize Simplification Opportunity: Recognize that the integral can be simplified by dividing the numerator by the denominator. $\newline$ $I = \int\frac{x-3}{x+3}\,dx$ $\newline$ We can perform long division or notice that $\frac{x-3}{x+3}$ can be rewritten as $1 - \frac{6}{x+3}$ .
Rewrite in Simplified Form: Rewrite the integral in terms of the simplified expression. $I = \int(1 - \frac{6}{x+3})\,dx$ This separates the integral into two simpler integrals.
Integrate First Term: Integrate the first term, which is the integral of $1$ with respect to $x$ . $I = \int 1\,dx - \int \left(\frac{6}{x+3}\right)dx$ The integral of $1$ with respect to $x$ is $x$ .
Integrate Second Term: Integrate the second term, which is the integral of $\frac{6}{x+3}$ with respect to $x$ .
$I = x - 6\int\left(\frac{1}{x+3}\right)dx$
The integral of $\frac{1}{x+3}$ with respect to $x$ is $\ln|x+3|$ .
Combine and Add Constant: Combine the results of the integrals and include the constant of integration. $\newline$ $I = x - 6\ln|x+3| + C$ $\newline$ This is the indefinite integral of the given function.