JEE Mains · Maths · STD 12 - 7.2 definite integral
If \(I _{ n }=\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \cot ^{ n } x dx ,\) then :
- A \(\frac{1}{ I _{2}+ I _{4}}, \frac{1}{ I _{3}+ I _{5}}, \frac{1}{ I _{4}+ I _{6}}\) are in \(G.P.\)
- B \(I _{2}+ I _{4}, I _{3}+ I _{5}, I _{4}+ I _{6}\) are in \(A.P.\)
- C \(I _{2}+ I _{4},\left( I _{3}+ I _{5}\right)^{2}, I _{4}+ I _{6}\) are in \(G.P.\)
- D \(\frac{1}{ I _{2}+ I _{4}}, \frac{1}{ I _{3}+ I _{5}}, \frac{1}{ I _{4}+ I _{6}}\) are in \(A.P.\)
Answer & Solution
Correct Answer
(D) \(\frac{1}{ I _{2}+ I _{4}}, \frac{1}{ I _{3}+ I _{5}}, \frac{1}{ I _{4}+ I _{6}}\) are in \(A.P.\)
Step-by-step Solution
Detailed explanation
\(I_{n}=\int_{\pi / 4}^{\pi / 2} \cot ^{n} x d x=\int_{\pi / 4}^{\pi / 2} \cot ^{n-2} x\left(\operatorname{cosec}^{2} x-1\right) d x\) \(\left.=-\frac{\cot ^{n-1} x}{n-1}\right]_{\pi / 4}^{\pi / 2}-I_{n-2}\) \(=\frac{1}{n-1}-I_{n-2}\)…
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