Match List-I with List-II (Consider _e x = x), List-I, List-II, Integral, Solution: where C is an arbitrary constant, (A) (f'(x))/(f(x) f(x)) dx, (I) x( x) + c, (B) (( x) + 1/( x)) dx, (II) x/( x) + C, (C) ( x)/((1+ x)^2) dx, (III) (f(x)) + c, (D) (1/( x) - 1/(( x)^2)) dx, (IV) x/(1+ x) + c, Choose the correct answer from the options given below:

Question

Match List-I with List-II (Consider $$\log_e x = \log x$$) | List-I | List-II | |---|---| | **Integral** | **Solution: where C is an arbitrary constant** | | (A) $$\displaystyle\int \dfrac{f'(x)}{f(x)\log f(x)}\,dx$$ | (I) $$x\log(\log x) + c$$ | | (B) $$\displaystyle\int \left(\log(\log x) + \dfrac{1}{\log x}\right) dx$$ | (II) $$\dfrac{x}{\log x} + C$$ | | (C) $$\displaystyle\int \dfrac{\log x}{(1+\log x)^2}\,dx$$ | (III) $$\log\left\vert \log(f(x)) \right\vert + c$$ | | (D) $$\displaystyle\int \left(\dfrac{1}{\log x} - \dfrac{1}{(\log x)^2}\right) dx$$ | (IV) $$\dfrac{x}{1+\log x} + c$$ | Choose the correct answer from the options given below:

Accepted Answer

(A) - (III), (B) - (I), (C) - (IV), (D) - (II)

Answer

(A) - (IV), (B) - (III), (C) - (II), (D) - (I)

Answer

(A) - (I), (B) - (II), (C) - (IV), (D) - (III)

Answer

(A) - (III), (B) - (II), (C) - (IV), (D) - (I)

List-I	List-II
Integral	Solution: where C is an arbitrary constant
(A) $\displaystyle\int \dfrac{f'(x)}{f(x)\log f(x)}\,dx$	(I) $x\log(\log x) + c$
(B) $\displaystyle\int \left(\log(\log x) + \dfrac{1}{\log x}\right) dx$	(II) $\dfrac{x}{\log x} + C$
(C) $\displaystyle\int \dfrac{\log x}{(1+\log x)^2}\,dx$	(III) $\log\left\vert \log(f(x)) \right\vert + c$
(D) $\displaystyle\int \left(\dfrac{1}{\log x} - \dfrac{1}{(\log x)^2}\right) dx$	(IV) $\dfrac{x}{1+\log x} + c$

List-I	List-II
(A) $\int \tan x \, dx$	(I) $\log\vert \sec x + \tan x\vert + c$
(B) $\int \cot x \, dx$	(II) $\log\vert \sec x\vert + c$
(C) $\int \sec x \, dx$	(III) $\log\vert \sin x\vert + c$
(D) $\int \cosec x \, dx$	(IV) $\log\vert \cosec x - \cot x\vert + c$

Solution

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