Consider the following Assertion (A) and Reas | vedclass

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(A) Step $1$: Analyze the Reason $(R)$. We know that $\sin^{-1}(u) + \cos^{-1}(u) = \frac{\pi}{2}$ for $u \in [-1, 1]$. Thus, the Reason $(R)$ is true

Vedclass · Accepted Answer

Both $(A)$ and $(R)$ are true, $(R)$ is the correct explanation of $(A)$

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(A) Step $1$: Analyze the Reason $(R)$. We know that $\sin^{-1}(u) + \cos^{-1}(u) = \frac{\pi}{2}$ for $u \in [-1, 1]$. Thus, the Reason $(R)$ is true.Step $2$: Simplify the integrand in Assertion $(A)$. Using the identity from $(R)$, $\sin^{-1}(\log x) + \cos^{-1}(\log x) = \frac{\pi}{2}$ provided $|\log x| \le 1$, i.e.,$x \in [1/e, e]$.Step $3$: Evaluate the integral: $\int \sqrt{x-3} \cdot \frac{\pi}{2} dx = \frac{\pi}{2} \int (x-3)^{1/2} dx$.Step $4$: Using the power rule $\int u^n du = \frac{u^{n+1}}{n+1}$, we get $\frac{\pi}{2} \cdot \frac{(x-3)^{3/2}}{3/2} + c = \frac{\pi}{2} \cdot \frac{2}{3}(x-3)^{3/2} + c = \frac{\pi}{3}(x-3)^{3/2} + c$.Step $5$: Since the result matches the Assertion $(A)$, $(A)$ is true and $(R)$ is the correct explanation.

Consider the following Assertion $(A)$ and Reason $(R)$:
Assertion $(A)$: $\int \sqrt{x-3} \left(\sin^{-1}(\log x) + \cos^{-1}(\log x)\right) dx = \frac{\pi}{3}(x-3)^{3/2} + c$
Reason $(R)$: $\sin^{-1}(f(x)) + \cos^{-1}(f(x)) = \frac{\pi}{2}$ for $|f(x)| \le 1$
Choose the correct option:

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Consider the following Assertion $(A)$ and Reason $(R)$:Assertion $(A)$: $\int \sqrt{x-3} \left(\sin^{-1}(\log x) + \cos^{-1}(\log x)\right) dx = \frac{\pi}{3}(x-3)^{3/2} + c$Reason $(R)$: $\sin^{-1}(f(x)) + \cos^{-1}(f(x)) = \frac{\pi}{2}$ for $|f(x)| \le 1$Choose the correct option: