_ ^ f'(x) [f(x)] ^2 ;dx =

MCQ

$\int_{}^{} {\frac{{f'(x)}}{{{{[f(x)]}^2}}}} \;dx = $

✓
$ - {[f(x)]^{ - 1}} + c$
B
$\log [f(x)] + c$
C
${e^{f(x)}} + c$
D
None of these

✓

Answer

Correct option: A.

$ - {[f(x)]^{ - 1}} + c$

a
(a) Put $f(x) = t \Rightarrow f'(x)\,dx = dt,$ then
$\int_{}^{} {\frac{{f'(x)}}{{|f(x){|^2}}}\,dx} = \int_{}^{} {\frac{1}{{{t^2}}}\,dt} = - \frac{1}{t} + c = - \frac{1}{{f(x)}} + c.$

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