F(x)=e^sin(x) +17 is an antiderivative of f(x)=e^sin(x)cos(x).
To determine if F(x) = e^sin(x) + 17 is indeed an antiderivative of f(x) = e^sin(x)cos(x), we can differentiate F(x) and see if it matches f(x)
To determine if F(x) = e^sin(x) + 17 is indeed an antiderivative of f(x) = e^sin(x)cos(x), we can differentiate F(x) and see if it matches f(x).
Let’s start by finding the derivative of F(x):
F'(x) = d/dx (e^sin(x) + 17)
To differentiate e^sin(x), we can use the chain rule:
d/dx (e^u) = e^u * du/dx,
where u = sin(x).
Therefore, d/dx (e^sin(x)) = e^sin(x) * cos(x).
Now, differentiating the constant term 17:
d/dx (17) = 0.
Adding the two derivatives together, we have:
F'(x) = e^sin(x) * cos(x) + 0
Simplifying, we get:
F'(x) = e^sin(x) * cos(x),
which is indeed the same as f(x).
Hence, F(x) = e^sin(x) + 17 is an antiderivative of f(x) = e^sin(x)cos(x).
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