Assertion (A): ₂^{e} ≤ft( 1/_e x - 1/(_e x)² ) dx = e - 2₂ e Reason (R): _{a}^{b} e^{x} (f(x)+f'(x)) dx = e^{b} f(b) - e^{a} f(a)

Correct Answer is : (A) and (R) are true, (R) is the correct explanation to (A).

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

(A) and (R) are false, (R) is not the correct explanation to (A).

(A) is true and (R) is false, R is not the correct explanation to (A).

(A) is false and (R) is true, (R) is not the correct explanation to (A).

Correct Answer is : (A) and (R) are true, (R) is the correct explanation to (A).

Test Index

AP EAPCET 06-Jul-2022 Shift 2 Paper

Section: Mathematics

Question : 74 of 160

Marks: +1, -0

Assertion (A):

\int\limits_{2}^{e} \left( \frac{1}{\log_e x} - \frac{1}{(\log_e x)^2} \right) dx = e - 2\log_2 e

\int\limits_{a}^{b} e^{x} (f(x)+f'(x)) dx = e^{b} f(b) - e^{a} f(a)

Solution: 👈: Video Solution

Assertion

Let

I = \int\limits_{2}^{e} \left( \frac{1}{\log_e x} - \frac{1}{(\log_e x)^2} \right) dx

Let

\log_e x = y

\;\Rightarrow\;\; x = e^{y}

\;\Rightarrow\;\; dx = e^{y} dy

I = \int\limits_{\log_e 2}^{1} e^{y} \left[ \frac{1}{y} + \left( \frac{-1}{y^2} \right) \right] dy

[Using the formula

\int\limits_{a}^{b} e^{x} [f(x)+f'(x)] dx = \left[ e^{x} f(x) \right] _a^b ]

I\; = \left[ e^{y} \; \frac{1}{y} \right]_{-\log e^2}^{1}

\; = e - \frac{2}{\log_e 2} = e - 2\log_2 e

Hence, Assertion and Reason both are true and Reason is the correct explanation of Assertion.

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