The value of ∫ dx/(1 + tan^3 x) (for x → 0,pi/2) is
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Indeed this was a Putnam question. The 2–√2 is pretty much irrelevant. Notice
I=∫π0dx1+tan(x)2√=∫0πd(π2−u)1+(tan(π2−u))2√=∫π0du1+cot(u)2√I=∫0πdx1+tan(x)2=∫π0d(π2−u)1+(tan(π2−u))2=∫0πdu1+cot(u)2
and
11+tan(x)2√=cos(x)
I=∫π0dx1+tan(x)2√=∫0πd(π2−u)1+(tan(π2−u))2√=∫π0du1+cot(u)2√I=∫0πdx1+tan(x)2=∫π0d(π2−u)1+(tan(π2−u))2=∫0πdu1+cot(u)2
and
11+tan(x)2√=cos(x)
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