Question

Consider the following
Assertion (A): $\int \sqrt{x-3}(\sin^{-1}(\log x) + \cos^{-1}(\log x))dx = \frac{\pi}{3}(x-3)^{3/2}+c$
Reason (R): $\sin^{-1}(f(x))+\cos^{-1}(f(x))=\frac{\pi}{2}$, $|f(x)|\le 1$
The correct answer is

• A. Both (A) and (R) are true, (R) is the correct explanation of (A)
• B. Both (A) and (R) are true, (R) is not the correct explanation of (A)
• C. (A) is true, but (R) is false
• D. (A) is false, but (R) is true

Hint:

When evaluating an assertion involving an integral or a function, always check the domain of definition first. If the domain is empty, any statement about the function's value (other than possibly zero for a definite integral) is likely false.

Answer 1

The Correct Option is D

Step 1: Analyze Reason (R).
The statement $\sin^{-1}(u)+\cos^{-1}(u) = \frac{\pi}{2}$ is a standard identity in inverse trigonometric functions. The condition for this identity to be valid is that $u$ must be in the domain of both $\sin^{-1}$ and $\cos^{-1}$, which is the interval $[-1, 1]$. So, the reason is true if $|u| \le 1$. The reason states this correctly with $u=f(x)$. Therefore, Reason (R) is a true statement.

Step 2: Analyze Assertion (A) by checking the domain of the integrand.
The integrand contains two parts. The term $\sqrt{x-3}$ requires its argument to be non-negative: $x-3 \ge 0 \implies x \ge 3$. The term $(\sin^{-1}(\log x) + \cos^{-1}(\log x))$ is equal to $\pi/2$, but only if the condition from Reason (R) is met. Here, the argument is $\log x$. So we require $|\log x| \le 1$. This means $-1 \le \log x \le 1$. Exponentiating with base $e$, we get $e^{-1} \le x \le e^1$, which is $\frac{1}{e} \le x \le e$.

Step 3: Find the overall domain of the integrand.
For the integral to be well-defined, $x$ must be in the intersection of the domains of both parts. Domain 1: $x \ge 3$. Domain 2: $\frac{1}{e} \le x \le e$. We know that $e \approx 2.718$. Therefore, the second domain is approximately $[0.368, 2.718]$. The intersection of $[3, \infty)$ and $[\frac{1}{e}, e]$ is the empty set, $\emptyset$.

Step 4: Conclude about Assertion (A).
Since the domain over which the integration is performed is empty, the integral is not well-defined in the way presented. An integral over an empty set is typically defined as 0. The assertion claims the integral is equal to $\frac{\pi}{3}(x-3)^{3/2}+c$, which is a non-zero function of $x$. Because the premise (the integral itself) is not valid for any real number $x$, the statement asserting its value is false. Therefore, Assertion (A) is false.

Step 5: Final Conclusion.
Reason (R) is a true statement, but Assertion (A) is a false statement. This corresponds to option (D).