Re: [積分] 絕對值不定積分
※ 引述《berry17523 (straw)》之銘言：
: 1. ∫∣x∣dx
: 2. ∫x∣x∣dx
: 第一題之前有人問過 答案是( x∣x∣) / 2
: 可是我不懂為什麼是這個答案...
: 不好意思喔 可以幫我解答嗎
Consider
x
∫ ∣t∣ dt, called (A)
0
As x≧0, (A) ＝ x^2/2
As x＜0, (A) ＝ -x^2/2
So, for any x, we have (A) ＝ x|x|/2.
In addition,
x a x
∫ ∣t∣ dt ＝ ∫ ∣t∣ dt ＋ ∫ ∣t∣ dt.
0 0 a
a
Say ∫ ∣t∣ dt ＝ - C.
0
x
That is, ∫ ∣t∣ dt ＝ (A) ＋ C ＝ x|x|/2 ＋ C.
a
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※ 編輯: math1209 來自: 114.32.219.116 (06/14 22:37)
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