 # Some Properties of Definite Integrals

• \int_{a}^{b} f(x) dx = - \int_{b}^{a} f(x) dx
• \int_{o}^{a} f(x) dx = \int_{o}^{a/2} f(x) dx + \int_{o}^{a/2} f(a-x) dx
• \int_{a}^{b} f(x) dx = \int_{a}^{b} f(a+b - x) dx
• \int_{o}^{a} f(x) dx = \int_{o}^{a} f(a - x) dx
• \int_{a}^{b} f(x) dx = 0 if f(a + x) = − f(b − x)
• \int_{a}^{b} f(x) dx = 2\int_{o}^{\frac{a+b}{2}} f(x) dx if f(a + x) = f(b − x)
• \int_{a}^{b} x f(x) dx = \frac{a+b}{2} \int_{a}^{b} f(x) dx if f(a + b - x) = f(x)
• \int_{-a}^{a} f(x) dx= \begin{cases}2 \int_{o}^{a} f(x) dx & ; \ if \ f(x) \ is \ even \\ o & ; \ if \ f(x) \ is \ odd \end{cases}
• \int_{o}^{2a} f(x) dx = \begin{cases}2 \int_{o}^{a} f(x) dx & ; \ if \ f(2a - x) = f(x) \\ o & ; \ if \ f(2a-x)= -f(x)\end{cases}
• \int_{o}^{a} x f(x) dx = \begin{cases} \frac{a}{2} \int_{o}^{a} f(x) dx & if \ f(a - x) = f(x) \\ a \int_{o}^{a/2} f(x) dx & if \ f(a-x) = f(x)\end{cases}

### Part2: View the Topic in this video From 00:40 To 13:13

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1. \int_{a}^{b} f(x) \ dx=\int_{a}^{b} f(t) \ dt

2. \int_{a}^{b} f(x) \ dx=-\int_{b}^{a} f(x) \ dx

3. \int_{a}^{b} f(x) \ dx=\int_{a}^{c} f(x) \ dx+\int_{c}^{b} f(x) \ dx, where a < c < b

4. \int_{a}^{a} f(x) \ dx=0

5. \int_{0}^{a} f(x) \ dx=\int_{0}^{a} f(a-x) \ dx
Deduction \int_{0}^{a} \frac{f(x)}{f(x)+f(a-x)} \ dx=\frac{a}{2}

6. \int_{a}^{b} f(x) \ dx=\int_{a}^{b} f(a+b-x) \ dx
Deduction \int_{a}^{b} \frac{f(x)}{f(x)+f(a+b-x)} \ dx=\frac{b-a}{2}

7. \int_{0}^{2a} f(x) \ dx=\int_{0}^{a} f(x) \ dx+\int_{0}^{a} f(2a-x) \ dx

8. \int_{-a}^{a} f(x) \ dx=\int_{0}^{a} f(x) \ dx+\int_{0}^{a} f(-x) \ dx

9. \int_{0}^{2a} f(x) \ dx=\begin{cases}2\int_{0}^{a} f(x) \ dx \ \ {\tt if}, & f(2a-x) = f(x)\\0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\tt if} & f(2a-x) = -f(x)\end{cases}

10. \int_{-a}^{a} f(x) \ dx=\begin{cases}2\int_{0}^{a} f(x) \ dx, & {\tt if} & f(x) \ is \ even \ i.e., f(-x)=f(x)\\0, & {\tt if} & f(x) \ is \ odd \ i.e., f(-x)=-f(x)\end{cases}

11. If \int_{a}^{b} f(x) \ dx=(b-a)\int_{0}^{1} f[(b-a)x + a] \ dx