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The following item consists of two statements, one labelled the Assertion $(A)$ and the other labelled the Reason $(R) .$ You are to examine these two statements carefully and decide if the Assertion (A) and Reason $(R)$ are individually true and if so, whether the reason is a correct explanation of the Assertion. Select your answer using the codes given below.
$Assertion (A)$: $\int \frac{e^{x}}{x}(1+x \log x) d x+c=e^{x} \log x$
$Reason$ $(\mathbf{R}): \int e^{x}\left[f(x)+f^{\prime}(x)\right] d x=e^{x} f(x)+c$
Options:
$Assertion (A)$: $\int \frac{e^{x}}{x}(1+x \log x) d x+c=e^{x} \log x$
$Reason$ $(\mathbf{R}): \int e^{x}\left[f(x)+f^{\prime}(x)\right] d x=e^{x} f(x)+c$
Solution:
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Verified Answer
The correct answer is:
Both A and $\mathrm{R}$ are true but $\mathrm{R}$ is not the correct explanation of $\mathrm{A}$
(A) Consider $\int \frac{e^{x}}{x}(1+x \log x) d x$
$=\int \frac{e^{x}}{x} d x+\int e^{x} \log x d x$
$=e^{x} \log x-\int e^{x} \log x d x+\int e^{x} \log x d x=e^{x} \log x$
(R) $\int e^{x}\left[f(x)+f^{\prime}(x)\right] d x$
$=\int e^{x} f(x) d x+\int e^{x} f^{\prime}(x) d x$
$=e^{x} f(x)-\int e^{x} f^{\prime}(x) d x+\int e^{x} f^{\prime}(x) d x=e^{x} f(x)+c$
Both Aand $\mathrm{R}$ are true but $\mathrm{R}$ is not the correct explanation of $\mathrm{A}$.
$=\int \frac{e^{x}}{x} d x+\int e^{x} \log x d x$
$=e^{x} \log x-\int e^{x} \log x d x+\int e^{x} \log x d x=e^{x} \log x$
(R) $\int e^{x}\left[f(x)+f^{\prime}(x)\right] d x$
$=\int e^{x} f(x) d x+\int e^{x} f^{\prime}(x) d x$
$=e^{x} f(x)-\int e^{x} f^{\prime}(x) d x+\int e^{x} f^{\prime}(x) d x=e^{x} f(x)+c$
Both Aand $\mathrm{R}$ are true but $\mathrm{R}$ is not the correct explanation of $\mathrm{A}$.
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