Why is there a "+C" term in the solution of every indefinite integral?
In order to understand where that annoying "+C" term comes from that is so easily forgotten from the end of the solution, let's start with the general thought process of solving an indefinite integral. When facing such a question, that is, "calculate the (indefinite) integral of this and that function", we think of the problem as follows. What is that function that if I differentiate I get the function I have to integrate? For example, let's say we want to find the indefinite integral of the function F(x) = x, which is a straight line. In our example the question then becomes "What is the function whose derivate is F(x) = x?".
If you have practised the rules of differentiation enough, then this is simple, right? If I differentiate G(x) = 0.5 * x2 I get G'(x) = 2 * 0.5 * x = x = F(x), which is our solution! But what if I differentiate G(x) = 0.5 * x2 + 1? I get G'(x) = 2 * 0.5 * x + 0 = x = F(x), which is also the solution! How about G(x) = 0.5 * x2 - 47? It leads to the same result. So how can we bring all the possible solutions to the question under one roof? This is where the "+C" term comes from: if we say that the solution is G(x) = 0.5 * x2 + C (where C is a constant), then our solution still remains G'(x) = F(x) = x, regardless whether C is 0 or -98549 or pi. In a nutshell, the "+C" term is there to express that we don't know what the exact is solution is, and writing it this way describes all the possible solutions to the question. Lastly, note that you don't have to stick to "+C"; you can use all kinds of notation instead as long as you make it clear that it is a constant.
