Q. 29 - 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.

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

Step-by-Step Solution

Answer

Ans:${\int }_b^{a}f\left(\mathrm{x}\right)d\mathrm{x}=-{\int }_{\mathrm{a}}^{\mathrm{b}}\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x}=F\left(a\right)-F\left(b\right)$

Step 1. Given information:

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

Step 2 .Solution:

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