티스토리 뷰

…을 가장한, 옛날에 AoPS에 답변으로 올렸지만 티스토리로 퍼오기 매우 귀찮아서 그냥 방치했던 계산 하나를 올려봅니다. 다른 꼴의 삼각함수 적분에도 쓸 수 있는 테크닉이 아닐까 해서 이렇게 올려봅니다.




Today's integral we are going to evaluate is a famous one,

.



Solution 1 (by elementary calculus). It is clear that . To determine for , we consider the difference for . Some trigonometric identities show that



Since



it follows that and for . Therefore for all .



Solution 2 (by complex analysis). It is easy to see, by the substitution , that

.

Thus for ,



Since



we have for .



Solution 3 (by advanced calculus). Let for . We further assume , for the convenience of calculation. Then



But note that



where is the harmonic number, and logarithmic differentiation of



gives



which implies that



as . Also, for , we find that



Therefore we have



which simplifies to



In special cases, taking for gives .
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