The fundamental theorem of calculus [Calculus] is a theorem that links the concept of differentiating [Differential calculus] a function (calculating its slopes, or rate of change at each time) with the concept of integrating [Integral calculus] a function (calculating the area under its graph, or the cumulative effect of small contributions). The two operations are inverses of each other apart from a constant value which depends on where one starts to compute area.
The first part of the theorem, the first fundamental theorem of calculus, states that for a function \(f\), an antiderivative [Antiderivative] or indefinite integral \(F\) may be obtained as the integral of \(f\) over an interval with a variable upper bound. This implies the existence of antiderivatives for continuous functions.
Conversely, the second part of the theorem, the second fundamental theorem of calculus, states that the integral of a function \(f\) over a fixed interval is equal to the change of any antiderivative \(F\) between the ends of the interval. This greatly simplifies the calculation of a definite integral provided an antiderivative can be found by symbolic integration, thus avoiding numerical integration.
First part
Given \(f\) is continuous over a Closed interval \([a,b]\):
\(F(x) = \int_a^x f(t)dt\), where \(x\) is in \([a,b]\)
$$
\begin{align} \frac{dF}{dx} &= \frac{d}{dx}\int_a^xf(t)dt = f(x) \frac{dF}{dx} &= f(x) \\ F’(x) &= f(x) \end{align}
$$
[paraphrased]
Example
\(\frac{d}{dx}(\int_{\pi}^x \frac{\href{/posts/cosine}{\cos}^2t}{\ln(t-\sqrt{t})}dt) = \frac{\href{/posts/cosine}{\cos}^2x}{\ln(x-\sqrt{x})}\)
Second part
Given:
- \(f\) is continuous over a Closed interval \([c,d]\)
- \(F(x) = \int_c^x f(t)dt\) by First fundamental theorem of calculus
- \(c < a < b < d\)
$$
\begin{align} \int_a^b f(t)dt &= \int_c^b f(t)dt - \int_c^a f(t)dt \\ &= F(b) - F(a) \end{align}
$$
[paraphrased]
(“Fundamental Theorem of Calculus” 2022), (Khan Academy 2013)