I have a question regarding integrating a indefinite integral, I keep on getting the incorrect answer, and I do not know how I would be able to acquire the right answer.
The question is : $\int \dfrac{x}{x+3} dx$
I keep getting the answer: $x+3 -3\ln(x+3)$
I apologise if this is a simple question.
Thank you
On
The indefinite integral is defined less to a constant, because it is a function whose derivative is the integrand function and the derivative of a constant is $0$.
So writing correctly your result you have the integral function: $F(x)=x+3-3\ln|x+3| +C$ and this is the same as $F(x)=x-3\ln|x+3|+C'$, with $3+C=C'$ , anyway a constant.
Since we have $$\frac{x}{x+3} = \frac{x+3 - 3}{x+3} = 1 - \frac{3}{x+3}$$ then it follows immediately that $$\bbox[10px, border: blue 1px solid]{\int \frac{x}{x+3} \, \mathrm{d}x = x - 3 \ln |x+3| + c}$$
This agrees with your answer since your extra constant term can simple be absorbed into your arbitrary constant, i.e: you can write $$x+3 -3 \ln |x+3| + c =x - 3 \ln |x+3| + d$$ where $d = c+3$.