निश्चित समाकलन के गुणधर्म:

• $\int_{a}^{b} f(x) dx = \int_{a}^{b} f(t) dt$

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

• $\int_{a}^{b} f(x) dx = \int_{a}^{c} f(x) dx + \int_{c}^{b} f(x) dx$

• $\int_{-a}^{a} f(x) dx = \int_{0}^{a} (f(x) + f(-x)) dx = \begin{cases} 2 \int_{0}^{a} f(x) dx, & \text{यदि } f(-x) = f(x) \\ 0, & \text{यदि } f(-x) = -f(x) \end{cases}$

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

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

• $\int_0^{2a} f(x) dx = \int_0^a (f(x) + f(2a-x)) dx = \begin{cases} 2 \int_0^a f(x) dx, & \text{यदि } f(2a-x) = f(x) \\ 0, & \text{यदि } f(2a-x) = -f(x) \end{cases}$

यदि $f(x)$ एक आवर्ती फलन है जिसकी आवृत्ति $T$ है, तो

• $\int_{0}^{nT} f(x) dx = n \int_{0}^{T} f(x) dx, \quad n \in \mathbb{Z}$

• $\int_{a}^{a+nT} f(x) dx = n \int_{0}^{T} f(x) dx, \quad n \in \mathbb{Z}, \quad a \in \mathbb{R}$

• $\int_{mT}^{nT} f(x) dx = (n-m) \int_{0}^{T} f(x) dx, \quad m, n \in \mathbb{Z}$

• $\int_{nT}^{a+nT} f(x) dx = \int_{0}^{a} f(x) dx, \quad n \in \mathbb{Z}, \quad a \in \mathbb{R}$

• $\int_{a+nT}^{b+nT} f(x) dx = \int_{a}^{b} f(x) dx, \quad n \in \mathbb{Z}, \quad a, b \in \mathbb{R}$

महत्वपूर्ण गुणधर्म:

• यदि $\psi(x) \leq f(x) \leq \phi(x)$ जहाँ $a \leq x \leq b$, तो $\int_{a}^{b} \psi(x) dx \leq \int_{a}^{b} f(x) dx \leq \int_{a}^{b} \phi(x) dx$

• यदि $m \leq f(x) \leq M$ जहाँ $a \leq x \leq b$, तो $m(b-a) \leq \int_{a}^{b} f(x) dx \leq M(b-a)$

• $\left|\int_{a}^{b} f(x) dx\right| \leq \int_{a}^{b} |f(x)| dx $

• यदि $f(x) \geq 0$ है $[a, b]$ पर तो $\int_{a}^{b} f(x) dx \geq 0$

लाइबनिट्ज़ प्रमेय:

$F(x) = \int_{g(x)}^{h(x)} f(t) dt \quad \Rightarrow \quad \frac{dF(x)}{dx} = h’(x) f(h(x)) - g’(x) f(g(x))$

योग की सीमा के रूप में निश्चित समाकल:

$\int_a^b f(x) dx = \lim_{n \rightarrow \infty} \sum_{r=0}^{n-1} h f(a+rh) = \lim_{n \rightarrow \infty} \sum_{r=0}^{n-1}\left(\frac{b-a}{n}\right) f\left(a+\frac{(b-a)r}{n}\right)$