…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 
.
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 thatSince
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. ThenBut note that
where
is the harmonic number, and logarithmic differentiation of gives
which implies that
as
. Also, for , we find thatTherefore we have
which simplifies to
In special cases, taking
for gives .
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아 , 이 어지러운 함수들이란 골치야 ㅋㅋ ㅠ
위에 초등 미적분학과 복소해석학을 이용한 방법은 이해가 가나 젤 밑의 증명방법은 엄청나네요...
그보다 이 문제를 어떻게 구하냐와 구한 문제를 어떻게 이렇게 손쉽게 풀 수 있는지 궁금함 [앞으로 자주 들릴듯 하네요]
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저도 잘하고 싶은데ㅠㅠㅠㅠㅠㅠㅠㅠㅠㅠ