If _ ^ f(x) ,dx = x e^ - |x| + f(x), then f(x) is - Vidyadip

MCQ

If $\int_{}^{} {f(x)\,dx} = x{e^{ - \log |x|}} + f(x),$ then $f(x)$ is

A
$1$
B
$0$
✓
$c{e^x}$
D
$\log x$

✓

Answer

Correct option: C.

$c{e^x}$

c
(c) $\int_{}^{} {f(x)dx = x{e^{\log \left| {\frac{1}{x}} \right|}} + f(x) \Rightarrow \int_{}^{} {f(x)dx = \frac{x}{{|x|}} + f(x)} } $

On differentiating both sides , we get $f(x) = 0 + f'(x)$

We know $\frac{d}{{dx}}({e^x}) = {e^x},\,\,$

$\therefore \,\,f(x) = c{e^x}$.

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