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Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is broken into two parts , which both basically mean the same thing. Firstly, given any differentiable function \(f(x)\), there is an anti-derivative of \(f'(x)\) to get back \(f(x)\). On the other hand, Given the anti-derivative of \(f(x)\), we can differentiate it to get back \(f(x)\)

For instance, \(f(x)=x^2\)

\(f'(x)=2x\)

\(\int f'(x)dx= \int 2xdx = x^2+C\)

When \(C=0\), \(\int f'(x)dx=x^2=f(x)\)

On the other hand, \(\int f(x)dx = \frac{1}{3} x^3+C\)

\((\int f(x) dx)'=(\frac{1}{3} x^3+C)'=x^2=f(x)\)

You are welcome to prove it below.

Note by Aloysius Ng
3 years, 1 month ago

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