[微積] 三題問題請教
臨時想不出那麼多題@@ 請板友幫忙一下這三題
pi/2
1. S (1+cosx)/(1+(cosx)^2)
0
idea:
我做了一下觀察不到什麼 wolfram跑出來會有arctanh...
然後看看他的不定積分...整個超扯超長 請問有什麼trick直接算此範圍定積分嗎
2. find indefinite integral of f^(-1)(y) , where f(x) = 1/(e^(4x)+e^(2x))
idea:
這種函數應該寫不出來f^(-1)(y)長怎樣
所以我用了一次分部積分後 (let g(y) = f^(-1)(y) )
得到 S g(y) dy = g(y) * y - S y*g'(y) dy
而我們知道 g'(y) = 1/f'(x) = 1/f'(g(y)) ... 然後我就投降了...
3. 用MVT證 for all x,y>0 ,(1+x)ln(1+x) + (1+y)ln(1+y) < (1+x+y)ln(1+x+y)
idea:
湊一段時間湊不出來= =...
謝謝幫忙
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