Q. 28 - Calculus

Question

The algebra of definite integrals:

Fill in the blanks to complete the definite integral rules that follow. You may assume that f and g are integrable functions on $[a,b]$, that $c\in [a,b]$, and that k is any real number.

• ${\int }_{\mathrm{a}}^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x}=\_\_\_\_\_\_.$
  

Step-by-Step Solution

Answer

Ans: ${\int }_{\mathrm{a}}^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x}=0$

Step 1. Given information:

${\int }_{\mathrm{a}}^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x}$

Step 2. Solution:

$$\begin{gathered} {\int }_{\mathrm{a}}^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x}={\int }_{\mathrm{a}}^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} \\ \mathrm{If} \mathrm{f} \mathrm{is} \mathrm{continuous} \mathrm{on} \left[\mathrm{a}\right] \mathrm{and} \mathrm{F} \mathrm{is} \mathrm{any} \mathrm{antiderivative} \mathrm{of} \mathrm{f} , \mathrm{then} \\ ={\left[\mathrm{F}\left(\mathrm{x}\right)\right]}_{\mathrm{a}}^{\mathrm{a}} \\ =\left(\mathrm{F}\left(\mathrm{a}\right)-\mathrm{F}\left(\mathrm{a}\right)\right) \\ =0 \end{gathered}$$