If f′(x)=g(x) for for a≤x≤b then ∫ab f(x)g(x)dx is equal to

∫ab f(x)g(x)dx=∫ab f′(x)f(x)dx=12(f(b))2−(f(a))2

If f′(x)=g(x) for for a≤x≤b then ∫ab f(x)g(x)dx is equal to

∫ab f(x)g(x)dx=∫ab f′(x)f(x)dx=12(f(b))2−(f(a))2

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If $f^{'} (x) = g (x)$ for $f o r a \leq x \leq b$ then $\int_{a}^{b} f (x) g (x) d x$ is equal to

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$f (b) - f (a)$

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

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

$g (b) - g (a)$

answer is C.

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Detailed Solution

$\begin{matrix} \int_{a}^{b} f (x) g (x) d x = \int_{a}^{b} f^{'} (x) f (x) d x \\ = \frac{1}{2} ((f (b))^{2} - (f (a))^{2}) \end{matrix}$

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If f′(x)=g(x) for for a≤x≤b then ∫ab f(x)g(x)dx is equal to

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If $f^{'} (x) = g (x)$ for $f o r a \leq x \leq b$ then $\int_{a}^{b} f (x) g (x) d x$ is equal to