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:


ANSWER

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


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
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