Mathematics

If $\dfrac{d}{dx}f(x) = g(x)$ for $a \le x \le b$, then $\overset{b}{\underset{a}{\displaystyle \int}}f(x)g(x)dx$ equals to:

$\dfrac{[f(b)]^2 - [f(a)]^2}{2}$

SOLUTION
Given $\dfrac{d}{dx}f(x) = g(x)$
To find $\displaystyle \int_{a}^{b}f(x)g(x)dx$
Sol : $\displaystyle \int_{a}^{b}f(x)g(x)dx$
Let $f(x) = t$
$\displaystyle \frac{d}{dx}f(x) = \frac{dt}{dx}$
$\displaystyle g(x) = \frac{dt}{dx}$
$\displaystyle g(x)dx = dt$
$\displaystyle \int f(x)g(x)dx = \int tdt = \frac{t^{2}}{2} = \frac{(f(x))^{2}}{2}$
evaluating above integral area the limits a to b ,
$\displaystyle \int_{a}^{b}f(x)g(x)dx = \frac{f(x)^{2}}{2}|_{a}^{b} = \frac{f(b)^{2}-f(a)^{2}}{2}$
$\displaystyle\therefore \int_{a}^{b}f(x)g(x)dx = \frac{f^{2}(b)-f^{2}(a)}{2}$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 105

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