Definite Integration

1. Definite Integral as Limit of Sum

Definition & Concept
The definite integral $\int_{a}^{b} f(x) dx$ is defined as the area of the region bounded by the curve $y=f(x)$, the X-axis, and the ordinates $x=a$ and $x=b$.

If $g(x)$ is the primitive (antiderivative) of $f(x)$, then the exact area of this region is given by $g(b) - g(a)$.
Limit of a Sum Formula
The interval $[a, b]$ can be divided into $n$ equal parts such that $a = x_0 < x_1 < x_2 < \dots < x_{n-1} < x_n = b$.

The area of each strip can be approximated by the area of a rectangle, which is $(x_{r+1} - x_r) \cdot f(t_r)$, where $t_r$ is a point on the curve between $x_r$ and $x_{r+1}$.

Taking the limit as $n \rightarrow \infty$, the sum of these rectangular areas gives the exact definite integral:

$$ \lim_{n \rightarrow \infty} \sum_{r=0}^{n-1} (x_{r+1} - x_r) \cdot f(t_r) = g(b) - g(a) = \int_{a}^{b} f(x) dx $$

Definite integral as the limit of a sum of rectangular strips

The Theorem
Let $f$ be a continuous function defined on $[a, b]$. If $\int f(x) dx = g(x) + c$, then the definite integral is evaluated as:

$$ \int_{a}^{b} f(x) dx = [g(x) + c]_{a}^{b} = g(b) - g(a) $$

In the expression $\int_{a}^{b} f(x) dx$, $a$ is called the lower limit and $b$ is called the upper limit.

Property I (Zero Interval):
$$ \int_{a}^{a} f(x) dx = 0 $$
Property II (Reversing Limits):
$$ \int_{a}^{b} f(x) dx = - \int_{b}^{a} f(x) dx $$
Property III (Dummy Variable):
Definite integration is independent of the variable.

$$ \int_{a}^{b} f(x) dx = \int_{a}^{b} f(t) dt $$
Property IV (Splitting the Interval):
$$ \int_{a}^{b} f(x) dx = \int_{a}^{c} f(x) dx + \int_{c}^{b} f(x) dx $$

Where $a < c < b$ (i.e., $c \in [a, b]$).
Property V (King's Rule):
$$ \int_{a}^{b} f(x) dx = \int_{a}^{b} f(a + b - x) dx $$
Property VI (Special Case of King's Rule):
$$ \int_{0}^{a} f(x) dx = \int_{0}^{a} f(a - x) dx $$
Property VII (Double Limit):
$$ \int_{0}^{2a} f(x) dx = \int_{0}^{a} f(x) dx + \int_{0}^{a} f(2a - x) dx $$
Property VIII (Even and Odd Functions):
If $f(x)$ is an EVEN function (i.e., $f(-x) = f(x)$):

$$ \int_{-a}^{a} f(x) dx = 2 \int_{0}^{a} f(x) dx $$

If $f(x)$ is an ODD function (i.e., $f(-x) = -f(x)$):

$$ \int_{-a}^{a} f(x) dx = 0 $$

Standard Identity
Derived from the definite integral properties (specifically King's Rule), the following standard identity holds true for general evaluation:

$$ \int_{a}^{b} \frac{f(x)}{f(x) + f(a + b - x)} dx = \frac{1}{2}(b - a) $$

Reduction Formula for Sine
To evaluate integrals of the type $\int_{0}^{\pi/2} \sin^n x dx$, specific reduction formulae are used based on whether $n$ is even or odd.

If $n$ is odd:

$$ \int_{0}^{\pi/2} \sin^n x dx = \frac{(n-1)}{n} \cdot \frac{(n-3)}{(n-2)} \cdot \frac{(n-5)}{(n-4)} \cdots \frac{4}{5} \cdot \frac{2}{3} $$

If $n$ is even:

$$ \int_{0}^{\pi/2} \sin^n x dx = \frac{(n-1)}{n} \cdot \frac{(n-3)}{(n-2)} \cdot \frac{(n-5)}{(n-4)} \cdots \frac{3}{4} \cdot \frac{1}{2} \cdot \frac{\pi}{2} $$
Equivalence for Cosine
By utilizing the definite integral property $\int_{0}^{a} f(x) dx = \int_{0}^{a} f(a - x) dx$, the reduction formula translates identically to cosine powers:

$$ \int_{0}^{\pi/2} \cos^n x dx = \int_{0}^{\pi/2} \left[\cos\left(\frac{\pi}{2} - x\right)\right]^n dx = \int_{0}^{\pi/2} \sin^n x dx $$