AP EAMCET · Maths · Indefinite Integration

If $f_n(x)=\log \log \log \ldots \log x \quad(\log$ is repeated $n$-times), then
$\int\left(x f_1(x) f_2(x) \ldots f_n(x)\right)^{-1} d x$ is equal to

A $f_{n+1}(x)+c$
B $\frac{f_{n+1}(x)}{n+1}+c$
C $n f_n(x)+c$
D $\frac{f_n(x)}{n}+c$

Verified Solution

Answer & Solution

Correct Answer

(A) $f_{n+1}(x)+c$

Step-by-step Solution

Detailed explanation

$f_n(x)=\log \cdot \log \cdot \log \ldots \cdot \log x$ (upto $n$ times) $f_1(x)=\log x$ $f_2(x)=\log \log x$ $f_3(x)=\log \log \log x$ $f_{n-1}(x)=\log \log \log \ldots \ldots \log x$ (up to $(n-1)$ times) Now,…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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From AP EAMCET

A body is projected horizontally from the top of a tower of height

180 m

with a velocity of

20 {ms}^{- 1}

. If acceleration due to gravity is

10 {ms}^{- 2}

then match the following.

List-I		List-II
$(A)$	Velocity of the body after $1$ second $($ in ${ms}^{- 1}$ )	(I)	$5$
$(B)$	Horizontal displacement of the body after $1$ second (in meters)	(ii)	$20$
$(C)$	Vertical displacement of the body after $1$ second (in meters)	(iii)	$10$
$(D)$	Vertical velocity of the body after $1$ second $($ in ${ms}^{- 1}$ )	(iv)	$22.4$

The correct answer is

AP EAMCET 2018 Medium

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If \(f_n(x)=\log \log \log \ldots \log x \quad(\log\) is repeated \(n\)-times), then
\(\int\left(x f_1(x) f_2(x) \ldots f_n(x)\right)^{-1} d x\) is equal to

Answer & Solution

Step-by-step Solution

See the Complete Solution

If \(f_n(x)=\log \log \log \ldots \log x \quad(\log\) is repeated \(n\)-times), then\(\int\left(x f_1(x) f_2(x) \ldots f_n(x)\right)^{-1} d x\) is equal to

Step-by-step Solution

If \(f_n(x)=\log \log \log \ldots \log x \quad(\log\) is repeated \(n\)-times), then
\(\int\left(x f_1(x) f_2(x) \ldots f_n(x)\right)^{-1} d x\) is equal to