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The following question consist of two statements, one labelled as the 'Assertion (A)' and the other as 'Reason (R)'. You are to examine these two statements carefully and select the answer.
Assertion $(\mathbf{A}): \int_{0}^{\pi} \sin ^{7} x d x=2 \int_{0}^{\pi / 2} \sin ^{7} x d x$
Reason(R) $: \sin ^{7} x$ is an odd function
Options:
Assertion $(\mathbf{A}): \int_{0}^{\pi} \sin ^{7} x d x=2 \int_{0}^{\pi / 2} \sin ^{7} x d x$
Reason(R) $: \sin ^{7} x$ is an odd function
Solution:
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Verified Answer
The correct answer is:
Both $\mathrm{A}$ and $\mathrm{R}$ are individually true but $\mathrm{R}$ is not the conect explamation A
$\int_{0}^{\pi} \sin ^{7} x d x=2 \int_{0}^{\pi / 2} \sin ^{7} x d x$
$\sin x$ is an odd function and for an odd function
$\int_{0}^{a} f(x) d x=2 \int_{0}^{a / 2} f(x) d x$
Hence, $\int_{0}^{\pi} \sin ^{7} x d x=2 \int_{0}^{\pi / 2} \sin ^{7} x d x$ is true.
So, A and $\mathrm{R}$ both are individually true but $\mathrm{R}$ is not the correct explanation of A.
$\sin x$ is an odd function and for an odd function
$\int_{0}^{a} f(x) d x=2 \int_{0}^{a / 2} f(x) d x$
Hence, $\int_{0}^{\pi} \sin ^{7} x d x=2 \int_{0}^{\pi / 2} \sin ^{7} x d x$ is true.
So, A and $\mathrm{R}$ both are individually true but $\mathrm{R}$ is not the correct explanation of A.
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