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Please,please help me - Integral Calculus - JEE Main-12

If F(x) = \ln xG(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)

 
Answers (1)
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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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