For the function f (x) = e ^ x + sinx, find the antiderivative f if you know that f (0) = – 1.

Let’s remember what an antiderivative is. The function F (x) is called the antiderivative for the function f (x) on the given interval, if for any x from the given interval F ‘(x) = f (x). We find the derivative in our case:
F ‘(x) = e ^ x – cos x + c;
we know that f (0) = -1, we substitute it into our expression:
– 1 = e ^ 0 – cos 0 + c;
c = -1 -1 +1 = -1;
So the antiderivative for the function:
F ‘(x) = e ^ x – cos x – 1.