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  4. If (d/d x) f(x)=g(x) for a ≤ x ≤ b, then ∫ limitsab f(x) g(x) d x equals to:

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Q. If $\frac{d}{d x} f(x)=g(x)$ for $a \leq x \leq b$, then $\int\limits_a^b f(x) g(x) d x$ equals to:

Integrals

Solution:

Given that
$ \frac{d}{d x} f(x)=g(x) $
$ I=\int\limits_a^b f(x) g(x) d x=\left[\frac{f^2(x)}{2}\right]_a^b $
$=\frac{f^2(b)-f^2(a)}{2}$