AP EAMCET · Maths · Definite Integration
Assertion (A): \(\int_{\frac{\pi}{2}}^{\frac{3 \pi}{2}}[2 \sin x] d x=0\), where [.] denotes the greatest integer function
Reason (R) : \(2 \sin x\) is a decreasing function in \(\left[\frac{\pi}{2}, \frac{3 \pi}{2}\right]\)
- A Both \(A\) and \(R\) are true and \(R\) is the correct explanation of \(\mathrm{A}\)
- B Both \(\mathrm{A}\) and \(\mathrm{R}\) are true but \(\mathrm{R}\) is not the correct explanation of \(A\)
- C \(\mathrm{A}\) is true, \(\mathrm{R}\) is false
- D \(\mathrm{A}\) is false, \(\mathrm{R}\) is true
Answer & Solution
Correct Answer
(D) \(\mathrm{A}\) is false, \(\mathrm{R}\) is true
Step-by-step Solution
Detailed explanation
\(\begin{aligned} & \text { (A) } \int_{\frac{\pi}{2}}^{\frac{3 \pi}{2}}[2 \sin x] d x=2 \int_{\frac{\pi}{2}}^{\frac{3 \pi}{2}}[\sin x] d x \\ & =2\left[\int_{\frac{\pi}{2}}^\pi 0 \mathrm{dx}+\int_\pi^{\frac{\pi}{2}}(-1) \mathrm{dx}\right] \\ & \end{aligned}\)…
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