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April 16th, 2015, 02:25 PM   #1
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Question The first fundamental theorem of calculus

Say I have the statement $\displaystyle \int \frac{\mathrm{d} y}{\mathrm{d} x}\mathrm{d}x = y$ How does the fundamental theorem of calculus make this necessarily true? When I see the formal statement of the theorem, it is usually in terms of a definite integral such as $\displaystyle F(x) = \int_{a}^{x}f(t)dt$. How does the later apply to the former if the former is an antiderivative and not a definite integral?
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April 16th, 2015, 04:06 PM   #2
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Originally Posted by Mr Davis 97 View Post
Say I have the statement $\displaystyle \int \frac{\mathrm{d} y}{\mathrm{d} x}\mathrm{d}x = y$ How does the fundamental theorem of calculus make this necessarily true? When I see the formal statement of the theorem, it is usually in terms of a definite integral such as $\displaystyle F(x) = \int_{a}^{x}f(t)dt$. How does the later apply to the former if the former is an antiderivative and not a definite integral?
It $\displaystyle doesn't$! What you should have is that $\displaystyle \int \frac{dy}{dx} dx= y+ C$ where "C" can be any constant.
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