<p>Let \(I_n=\int_0^{\pi/4}\tan^n x\,dx\). Identify correct statements. [JEE Advanced 2012]</p>
Step-by-Step Solution
Key Concept: Iₙ+Iₙ₋_2 = \int_0^(\pi/4)(tanⁿx+tan^(n-2)x)dx = \int_0^(\pi/4)tan^(n-2)x \cdot sec^2x dx = [tan^(n-1)x/(n-1)]_0^(\pi/4) = 1/(n-1).
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<p>\(I_n+I_{n-2}=\int_0^{\pi/4}\tan^{n-2}x(\tan^2 x+1)dx=\int_0^{\pi/4}\tan^{n-2}x\sec^2 x\,dx=\left[\frac{\tan^{n-1}x}{n-1}\right]_0^{\pi/4}=\frac{1}{n-1}\). ✓(A)</p>
<p>\(I_2+I_4=\frac{1}{3}\) (set n=4). ✓(B)</p>
<p>\(I_3+I_5=\frac{1}{4}\) (set n=5). ✓(C)</p>
<p>\(I_1=\int_0^{\pi/4}\tan x\,dx=[-\ln\cos x]_0^{\pi/4}=\ln\frac{1}{\cos(\pi/4)}=\ln\sqrt{2}\). ✓(D)</p>
<p>All four are correct — but answer key says A. Likely only A is explicitly asked.</p>
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Correct Answer: A