If int {f(x),dx = f(x)} , then int {{{left[ { | vedclass

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(A) Given that $\int {f(x)\,dx = f(x)} $. Differentiating both sides with respect to $x$,we get: $\frac{d}{dx} \left( \int {f(x)\,dx} \right) = \frac{

Vedclass · Accepted Answer

$\frac{1}{2}{\left[ {f\left( x \right)} \right]^2}$

Vedclass · Answer

(A) Given that $\int {f(x)\,dx = f(x)} $. Differentiating both sides with respect to $x$,we get: $\frac{d}{dx} \left( \int {f(x)\,dx} \right) = \frac{d}{dx} f(x)$ $f(x) = f'(x)$. Now,we need to evaluate $\int {{{\left[ {f(x)} \right]}^2}\,dx}$. Since $f(x) = f'(x)$,we can write the integral as: $\int {{{\left[ {f(x)} \right]}^2}\,dx} = \int {f(x) \cdot f'(x)\,dx}$. Let $f(x) = u$,then $f'(x)\,dx = du$. The integral becomes $\int {u\,du} = \frac{u^2}{2} + C$. Substituting $u = f(x)$ back,we get $\frac{1}{2}{\left[ {f(x)} \right]^2} + C$. Thus,the correct option is $A$.

If $\int {f(x)\,dx = f(x)} ,$ then $\int {{{\left[ {f(x)} \right]}^2}\,dx}$ is

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