Are these two answers same?

Question

intergrate x/(x+1) dx

my answer
= x + 1 - ln(x+1) + C

answer from book:
= x - ln (x+1) + C

Answer 1

Yes, your answer = x + 1 - ln(x+1) + C = x - ln(x+1) + (1+C) = x - ln(x+1) + C' = answer from book

Answer 2

Yes, these two answers are the same because in your answer [ x + 1 - ln(x+1) + C] the 1 will cancel with the C since it is just a constant and C stands for any constant number. For example, C = C + 1 = C + 100.

Answer 3

You could say they are the "same" because you can write your answer like this:
x - ln(x+1) + (1+C)
which is x - ln(x+1) "plus a constant". It's just not the same exact constant "C" in both cases.

This is why it's important to add the "+ C" on indefinite integrals. Taking the derivative of either makes the constants go away and gives you the same answer.

Answer 4

No, they are not the same. Your answer is correct.
In general,
integral(xdx/(bx+a) = 1/b^2[a+bx -aln(a+bx)] +C

In your case a=b=1 so your answer is right.

It would not be right to incude the 1 as part of the constant C.

Answer 5

Not the same