Properties of definite integral:

PYQ-2023-Definite_Integration-Q1 , PYQ-2023-Definite_Integration-Q2, PYQ-2023-Definite_Integration-Q5, PYQ-2023-Definite_Integration-Q9, PYQ-2023-Definite_Integration-Q10, PYQ-2023-Definite_Integration-Q16, PYQ-2023-Definite_Integration-Q18, PYQ-2023-Definite_Integration-Q20, PYQ-2023-Definite_Integration-Q23, PYQ-2023-Definite_Integration-Q24, PYQ-2023-AOD-Q12

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

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

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

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

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

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

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

If $f(x)$ is a periodic function with period $T$, then

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

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

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

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

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

Important Properties:

• If $\psi(x) \leq f(x) \leq \phi(x)$ for $a \leq x \leq b$, then $$\int_{a}^{b} \psi(x) dx \leq \int_{a}^{b} f(x) dx \leq \int_{a}^{b} \phi(x) dx$$

• If $m \leq f(x) \leq M$ for $a \leq x \leq b$, then $$m(b-a) \leq \int_{a}^{b} f(x) dx \leq M(b-a)$$

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

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

Leibnitz Theorem:

PYQ-2023-Definite_Integration-Q4, PYQ-2023-Definite_Integration-Q14, PYQ-2023-Definite_Integration-Q17, PYQ-2023-Definite_Integration-Q20

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

Definite Integrals As A Limit Of Sum:

PYQ-2023-Definite_Integration-Q15, PYQ-2023-Definite_Integration-Q21

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