Definite-integration Formula Sheet

Definite integrals compute the net area under a curve between two specified points, providing a numerical value representing the total accumulation of the function over that interval.
A definite integral is denoted by $\int_a^b f(x) dx$ which represents the algebraic area bounded by the curve y = f(x), the ordinates x = a, x = b and the x-axis.

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• 1. The Fundamental Theorem of Calculus
• 2. Properties of Definite Integral
• 3. Walli's Formula
• 4. Derivative of Antiderivative Function
• 5. Definite Integral as Limit of a Sum
• 6. Estimation of Definite Integral
• 7. Some Standard Results

1. The Fundamental Theorem of Calculus:

(a) If $f$ is continuous on $[a, b]$, then the function $g$ defined by:
$g(x) = \int_{a}^{x} f(t) \, dt, \quad a \leq x \leq b$
is continuous on $[a, b]$ and differentiable on $(a, b)$, and $g'(x) = f(x)$.

(b) If $f$ is continuous on $[a, b]$, then: br/> $\int_{a}^{b} f(x) \, dx = F(b) - F(a)$ br/> where $F$ is any antiderivative of $f$, that is, a function such that $F' = f$.

Note:
If $\int_{a}^{b} f(x) \, dx = 0$ then the equation $f(x) = 0$ has at least one root lying in $(a, b)$ provided $f$ is a continuous function in $(a, b)$.

2. Properties of Definite Integral:

$\bold{(a)}$ $\int_{a}^{b} f(x) \, dx = \int_{a}^{b} f(t) \, dt \Rightarrow \int_a^bf(x)dx$ the integral does not depend upon the variable of integration. It is a numerical quantity.

$\bold{(b)}$ $\int_{a}^{b} f(x) \, dx = -\int_{b}^{a} f(x) \, dx$

$\bold{(c)}$ $\int_{a}^{b} f(x) \, dx = \int_{a}^{c} f(x) \, dx + \int_{c}^{b} f(x) \, dx$
where $c$ may lie inside or outside the interval $[a, b]$. This property is used when $f$ is piecewise continuous in $(a, b)$.

$\bold{(d)}$ $\int_{-a}^{a} f(x)dx = \int_0^a[f(x) + f(-x)]dx = \begin{cases} 0 &\text{;if } f(x) \space is \space an \space odd \space function \\ 2\int_0^a f(x)dx &\text{;if } f(x) \space is \space an \space even \space function \end{cases}$

$\bold{(e)}$ $\int_{a}^{b} f(x) \, dx = \int_{a}^{b} f(a + b - x) \, dx$ In particular, $\int_{0}^{a} f(x) \, dx = \int_{0}^{a} f(a - x) \, dx$

$\bold{(f)}$ $\int_{0}^{2a} f(x) \, dx = \int_{0}^{a} f(x) \, dx + \int_{0}^{a} f(2a - x) \, dx = \begin{cases} 2\int_a^b f(x)dx &\text{;if } f(2a-x) = f(x) \\ 0 &\text{;if } f(2a-x) = -f(x) \end{cases}$

$\bold{(g)}$ $\int_{0}^{nT} f(x) \, dx = n\int_{0}^{T} f(x)dx, (n \in I);$ where ‘T’ is the period of the function i.e. f(T + x) = f(x)

Note: $\int_{x}^{T + x} f(t) \, dt$ will be independent of $x$ and equal to $\int_{0}^{T} f(t) dt$

$\bold{(h)}$ $\int_{a+nt}^{b+nT} f(x)dx = n \int_{a}^{b} f(x)dx$ where $f$ is periodic with period $T$ and $n \in \mathbb{I}$.

$\bold{(i)}$ $\int_{ma}^{na} f(x) \, dx = (n - m) \int_{0}^{a} f(x)dx , (n,m \in I)$ if $f(x)$ is periodic with period 'a'.

3. Walli's Formula:

$\bold{(a)}$ $\int_{0}^{\pi/2} \sin^n x \, dx = \int_{0}^{\pi/2} \cos^n x \, dx = \frac{(n-1)(n-3)\ldots(1 \text{ or } 2)}{n(n-2)\ldots(1 \text{ or } 2)}k$
Where k = $= \begin{cases} \frac{\pi}{2} &\text{if } n \, is \, even \\ 1 &\text{if } n \, is \, odd \end{cases}$

$\bold{(b)}$ $\int_{0}^{\pi/2} \sin^n x \cos^m x \, dx = \frac{[(m-1)(m-3)\ldots(1 \text{ or } 2)][(n-1)(n-3)\ldots(1 \text{ or } 2)]}{(m+n)(m+n-2)\ldots(1 \text{ or } 2)}$
Where k = $= \begin{cases} \frac{\pi}{2} &\text{if } m, n \, are \, even (m,n \in N) \\ 1 &\text{if } otherwise \end{cases}$

4. Derivative of Antiderivative Function (Newton-Leibnitz Formula):

If $h(x)$ and $g(x)$ are differentiable functions of $x$ then,

$\frac{d}{dx} \left( \int_{g(x)}^{h(x)} f(t) \, dt \right) = f[h(x)] \cdot h'(x) - f[g(x)] \cdot g'(x)$

5. Definite Integral as Limit of a Sum:

$\int_{a}^{b} f(x) \, dx = \lim\limits_{n \to \infty} h \left[ f(a) + f(a + h) + f(a + 2h) + \ldots + f(a + \overline{(n-1)}h) \right] = \lim\limits_{n \to \infty} h \displaystyle\sum_{r=1}^{n-1} f(a+rh)$

where, $(b - a) = nh$.

If $a = 0$ and $b = 1$ then, $\lim\limits_{n \to \infty} h \displaystyle\sum_{r=0}^{n-1} f(rh) = \int_{0}^{1} f(x) \, dx$
where nh = 1

OR
$\lim\limits_{n \to \infty} (\frac{1}{n}) \displaystyle\sum_{r=1}^{n-1} f(\frac{r}{n}) = \int_{0}^{1} f(x) \, dx$

6. Estimation of Definite Integral:

$\bold{(a)}$ If $f(x)$ is continuous in $[a, b]$ and its range in this interval is $[m, M]$, then
$m(b - a) \leq \int_{a}^{b} f(x) \, dx \leq M(b - a)$

$\bold{(b)}$ If $f(x) \leq g(x)$ for $a \leq x \leq b$ then
$\int_{a}^{b} f(x) \, dx \leq \int_{a}^{b} g(x) \, dx$

$\bold{(c)}$ $\int_{a}^{b} |f(x)| \, dx \geq \left| \int_{a}^{b} f(x) \, dx \right|$

$\bold{(d)}$ If $f(x) \geq 0$ on the interval $[a,b]$, then $\int_{a}^{b} f(x) \, dx \geq 0$

$\bold{(e)}$ If $f(x)$ and $g(x)$ are two continuous functions on $[a, b]$ then
$|\int_{a}^{b} f(x)g(x) \, dx| \leq \sqrt{\left( \int_{a}^{b} f^2(x) \, dx \right) \left( \int_{a}^{b} g^2(x) \, dx \right)}$

7. Some Standard Results:

$\bold{(a)}$ $\int_{0}^{\pi/2} \log(\sin x) \, dx = \int_{0}^{\pi/2} \log(\cos x) \, dx = -\frac{\pi}{2} \log 2$

$\bold{(b)}$ $\int_{a}^{b} \{x\} \, dx = \frac{b - a}{2}, \quad a, b \in \mathbb{I}$

$\bold{(c)}$ $\int_{a}^{b} \frac{1}{x} \, dx = \log \left| \frac{b}{a} \right|$

$\bold{(d)}$ $\int_{a}^{b} \frac{|x|}{x} \, dx = |b| - |a|$

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