4. Fundamental Theorem of Calculus

Fundamental Theorem V.1

Theorem

If the function $f(x)$ is integrable* over the interval $[a,b]$, then

${\int }_a^{b}f(x)dx=F(b)-F(a)$

where $F(x)$ is any anti-derivative of $f(x)$.

Definition

A function $F(x)$ is an antiderivative of $f(x)$ if $\frac{dF}{dx}=f(x)$.

Anti-derivative

Definition

A function $F(x)$ is an antiderivative of $f(x)$ if $\frac{dF}{dx}=f(x)$.

Example

$F(x)=\frac{1}{3}{x}^3-\cos (x)$ is an anti-derivative of $f(x)=x^2+\sin (x)$.

Indeed

$F^{\prime }(x)=f(x)$

Fundamental Theorem V.2

Theorem

If the function $f(x)$ is integrable* over the interval $[a,b]$, then

$F(x)={\int }_a^{x}f(t)dt$

is an anti-derivative of $f(x)$, i.e. $F^{\prime }(x)=f(x)$.

Example

${\int }_0^{\frac{\pi }{2}}\cos (x)dx=F(\frac{\pi }{2})-F(0)$

where $F^{\prime }(x)=\cos (x)$

So

${\int }_0^{\frac{\pi }{2}}\cos (x)dx=\sin (\frac{\pi }{2})-\sin (0)=1-0=1$

since $\frac{d\sin (x)}{dx}=\cos (x)$

Elementary Functions

functions	Anti-derivatives
$x^r$ for $r\ne -1$	$\frac{x^r}{r+1}$
$x^{-1}=\frac{1}{x}$	$\ln (x)$
$\sin (x)$	$\cos (x)$
$\cos (x)$	$-\sin (x)$
$e^x$	$e^x$

Combinations

Elementary Functions:

• Polynomials
• Trigonometric Functions
• Exponential Function

Examples of Combination:

• $f(x)=x^2{e}^x$
• $g(x)=\cos (x^2)$

Problematic Functions

The anti-derivative $F(x)$ of $\cos (x^2)$ can only be expressed as:

$F(x)={\int }_a^x\cos (t^2)dt$

It can not be expressed as a combination of elementary functions.

Indefinite Integral

Notation

Given an integrable function $f(x)$, the indefinite integral

$\int f(x)dx=F(x)+C$

denotes the family of anti-derivative $F(x)+C$ where $F^{\prime }(x)=f(x)$ and $C$ represents an arbitrary constant.