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If $F(x) = \ln x$$G(x) = \ln \frac{x}{2}$$H(x) = \ln \frac{x}{e}$ and $f(x) = F'(x)$$g(x) = G'(x)$ and $h(x) =H'(x)$. Then

• Option 1)

$f(x) > g(x) > h(x)$

• Option 2)

$f(x) < g(x) < h(x)$

• Option 3)

$f(x) = g(x) = h(x)$

• Option 4)

$f(x) < h(x) < g(x)$

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As we have learnt,

Reason for indefinite integration -

We know that   $\frac{\mathrm{d} }{\mathrm{d} x}\left ( c \right )=0$  , this implies that  $F\left ( x \right )$  and $F\left ( x \right )+c$  are both integrals of the same function $f\left ( x \right )$ . For different values of $c$, we obtain different integrals of $f\left ( x \right )$. So $f\left ( x \right )$ is not definite hence indefinite .

- wherein

Where   $\frac{\mathrm{d}}{\mathrm{d}x}\left ( c \right )$ is differential of constant w.r.to $x$

$F'(x) = \frac{1}{x}$

Simillarly,

$G'(x) = \frac{1}{x} = H'(x)$

Option 1)

$f(x) > g(x) > h(x)$

Option 2)

$f(x) < g(x) < h(x)$

Option 3)

$f(x) = g(x) = h(x)$

Option 4)

$f(x) < h(x) < g(x)$

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