indefinite integration

👨🏻‍🏫 學術台數學

Larry

2011-03-27 7:07 pm

∫x^3/(1+4x^2)dx

更新1:

the ans is (1/8)x^2 - (1/16)ln(1+4x^2)^(1/2) + C

更新2:

無野la 自由自在的ans is correct

回答 (2)

myisland8132

2011-03-27 7:37 pm

✔ 最佳答案

Let 2x = tanθ, 2dx = (secθ)^2 dθ

∫ x^3/(1 + 4x^2) dx

= ∫ [(tanθ)^3/8] /(secθ)^2 * [(secθ)^2/2] dθ

= ∫ (tanθ)^3/16 dθ

= (1/32)(secθ)^2 + (1/16)ln|cosθ| + C (where C is a constant)

= (1/32)(1 + 4x^2) + (1/16)ln|1/√(1 + 4x^2)| + C

= (1/32)[(1 + 4x^2) - ln|1 + 4x^2|] + C

參考: http://www.wolfram.com/mathematica/

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用戶9112

2011-03-27 7:21 pm

http://i1090.photobucket.com/albums/i376/Nelson_Yu/1-164.jpg

圖片參考：http://i1090.photobucket.com/albums/i376/Nelson_Yu/1-164.jpg

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