If f(x) ,dx = f(x) , then [ f(x) ] ^2 , ,dx is - Vidyadip

MCQ

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

✓
$\frac{1}{2}{\left[ {f\left( x \right)} \right]^2}$
B
${\left[ {f\left( x \right)} \right]^3}$
C
$\frac{{{{\left[ {f\left( x \right)} \right]}^3}}}{3}$
D
${\left[ {\,f\left( x \right)} \right]^2}$

✓

Answer

Correct option: A.

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

a
(a) Trick : Let $f(x) = {e^x},$ then$\int {f(x)dx = f(x)} $
$\therefore \,\int {{{({e^x})}^2}dx = \frac{{{e^{2x}}}}{2} = \frac{1}{2}{{[f(x)]}^2}.} $

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