Fundamental Theorem of Calculus

The First Fundamental Theorem of Calculus (FTC) states that if \(f(x), F(x) \) are functions such that \(\int f(x)dx =F(x)+C\) and \(f(x)\) is continuous on the interval \([a,b]\) then \(\displaystyle\int_{a}^{b}f(x)dx=F(b)-F(a) \) whose evaluation is sometimes written as  follows

\(\displaystyle\int_{a}^{b}f(x)dx=F(x)\bigg|_a^b=F(b)-F(a)\)

It is IMPORTANT to note that if the integrated \(f(x)\) is discontinuous on the given interval \([a,b]\) then the integral is an improper integral and the FTC cannot be directly applied.

The Second Fundamental Theorem of Calculus (2^nd FTC) is a direct consequence of computing the derivative of integral by using the Chain Rule and the FTC.

If \(f(x), F(x) \) are functions such that \(\int f(x)dx =F(x)+C\) and \(f(x)\) is continuous on the interval \([a,b]\) then

\(\displaystyle\frac{d}{dx}\left(\int_{a}^{g(x)}f(t)dt\right)=\frac{d}{dx}\left(F(g(x))-F(a)\right) = F'(g(x))g'(x) – 0= f\big(g(x)\big)g'(x) \)

which can be generalized as follows

\(\displaystyle\frac{d}{dx}\left(\int_{h(x)}^{g(x)}f(t)dt\right)=f\big(g(x)\big)g'(x) -f\big(h(x)\big)h'(x)\)