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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
2 years, 5 months ago