除了常量不定积分是独一无二的

Let $f(x)$ be a function with antiderivative $F(x)$. We then know that $$\int f(x) dx=F(x).$$ However, this, in a sense, is partially incorrect. Let's see an example.

We are told to find the derivative of $F_1(x)=x^2$. This is simple differentiation, resulting in $f(x)=2x$. Now let's differentiate $F_2(x)=x^2+1$. It's $2x$ as well. But hey, if I told you to find the antiderivative of $f(x)=2x$, how could it be $F(x)=x^2+C$ for any value of $C$? Recall that the derivative of a constant is $0$, so essentially the constant does not count towards the derivative. Thus, we can use any constant.

Look at an integration table or other list of antiderivatives and watch for the additional constants. To specify the constant $C$, we need an additional condition, that we will use to solve for $C$.