Question

Assertion (A): \[ \int_{2}^{e}\left(\frac{1}{\log_e x}-\frac{1}{(\log_e x)^2}\right)\,dx = e-2\log_2 e \] Reason (R): \[ \int_{a}^{b} e^x\left(f(x)+f'(x)\right)\,dx = e^bf(b)-e^af(a) \]

• (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).

Hint:

Whenever an integrand looks like a combination of a function and its derivative, try to express it as the derivative of a product or quotient.

Answer 1

The Correct Option is A

Step 1: Observe the integrand in Assertion (A).
We have \[ \int_{2}^{e}\left(\frac{1}{\log_e x}-\frac{1}{(\log_e x)^2}\right)\,dx \] Since \[ \log_e x=\ln x, \] the integral becomes \[ \int_{2}^{e}\left(\frac{1}{\ln x}-\frac{1}{(\ln x)^2}\right)\,dx \]

Step 2: Identify the derivative form.
Consider \[ \frac{x}{\ln x} \] Differentiating, \[ \frac{d}{dx}\left(\frac{x}{\ln x}\right) = \frac{(\ln x)(1)-x\cdot \frac{1}{x}}{(\ln x)^2} \] \[ = \frac{\ln x-1}{(\ln x)^2} \] \[ = \frac{1}{\ln x}-\frac{1}{(\ln x)^2} \] So, \[ \frac{1}{\ln x}-\frac{1}{(\ln x)^2} = \frac{d}{dx}\left(\frac{x}{\ln x}\right) \]

Step 3: Evaluate the definite integral.
Thus, \[ \int_{2}^{e}\left(\frac{1}{\ln x}-\frac{1}{(\ln x)^2}\right)\,dx = \left[\frac{x}{\ln x}\right]_{2}^{e} \] \[ = \frac{e}{\ln e}-\frac{2}{\ln 2} \] Since \[ \ln e=1, \] we get \[ = e-\frac{2}{\ln 2} \] Also, \[ \log_2 e=\frac{1}{\ln 2} \] Therefore, \[ e-\frac{2}{\ln 2} = e-2\log_2 e \] Hence, Assertion (A) is true.

Step 4: Verify Reason (R).
Consider \[ \frac{d}{dx}\left(e^x f(x)\right) \] Using product rule, \[ \frac{d}{dx}\left(e^x f(x)\right) = e^x f(x)+e^x f'(x) \] \[ = e^x(f(x)+f'(x)) \] Therefore, \[ \int_a^b e^x(f(x)+f'(x))\,dx = \left[e^x f(x)\right]_a^b \] \[ = e^bf(b)-e^af(a) \] Hence, Reason (R) is also true.

Step 5: Check whether Reason explains Assertion.
The assertion follows from the same product-rule idea, because the integrand is the derivative of \[ \frac{x}{\ln x} \] Thus, Reason (R) gives the correct method behind Assertion (A).

Step 6: Final conclusion.
Therefore, \[ \boxed{\text{(A) and (R) are true, and (R) is the correct explanation to (A).}} \]