AP EAMCET · Maths · Definite Integration

Assertion (A)
\[
\int_2^e\left(\frac{1}{\log _e x}-\frac{1}{\left(\log _e x\right)^2}\right) d x=e-2 \log _2 e
\]
Reason (R)
\[
\int_a^b e^x\left(f(x)+f^{\prime}(x)\right) d x=e^b f(b)-e^a f(a)
\]

A A and \(R\) are true, \(R\) is the correct explanation to \(A\).
B \(A\) and \(R\) are false, \(R\) is not the correct explanation to \(\mathrm{A}\).
C \(A\) is true and \(R\) is false, \(R\) is not the correct explanation to \(\mathrm{A}\).
D \(\mathrm{A}\) is false and \(\mathrm{R}\) is true, \(\mathrm{R}\) is not the correct explanation to \(\mathrm{A}\).

Verified Solution

Answer & Solution

Correct Answer

(A) A and \(R\) are true, \(R\) is the correct explanation to \(A\).

Step-by-step Solution

Detailed explanation

Assertion Let \(I=\int_2^e\left(\frac{1}{\log _e x}-\frac{1}{\left(\log _e x\right)^2}\right) d x\) Let \(\log _e x=y\)…

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.

Same subject

Explore more questions on app

From AP EAMCET

Explore more questions on app

Assertion (A)\[\int_2^e\left(\frac{1}{\log _e x}-\frac{1}{\left(\log _e x\right)^2}\right) d x=e-2 \log _2 e\]Reason (R)\[\int_a^b e^x\left(f(x)+f^{\prime}(x)\right) d x=e^b f(b)-e^a f(a)\]

Step-by-step Solution

Assertion (A)
\[
\int_2^e\left(\frac{1}{\log _e x}-\frac{1}{\left(\log _e x\right)^2}\right) d x=e-2 \log _2 e
\]
Reason (R)
\[
\int_a^b e^x\left(f(x)+f^{\prime}(x)\right) d x=e^b f(b)-e^a f(a)
\]