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

Question

If (d/d x) f(x)=g(x) for a ≤ x ≤ b, then ∫ limitsab f(x) g(x) d x equals to: (A) f(2)-f(1) (B) g(2)-g(1) (C) ([f(b)]2-[f(a)]2/2) (D) ([g(b)]2-[g(a)]2/2)

Tardigrade Admin · Accepted Answer

Given that (d/d x) f(x)=g(x) I=∫ limitsab f(x) g(x) d x=[(f2(x)/2)]ab =(f2(b)-f2(a)/2)

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

$f (2) - f (1)$

$g (2) - g (1)$

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

$\frac{[ g ( b ) ] ^{2} - [ g ( a ) ] ^{2}}{2}$

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

Q. If dxd​f(x)=g(x) for a≤x≤b, then a∫b​f(x)g(x)dx equals to:

f(2)−f(1)

g(2)−g(1)

2[f(b)]2−[f(a)]2​

2[g(b)]2−[g(a)]2​