Re: [考古] 86清大理工
※ 引述《monene5566 (機掰凱文)》之銘言：
: 囧 又有好幾題需要版友指教了
: 2.
: 1 x^2 + 1
: S ---------------- dx=?
: 0 x^4 + 1
: 試過許多替換 但都不行 請指教 謝謝:)
x^2 + 1
= ---------------
(x^2+1)^2-2x^2
x^2 + 1
=-------------------------
(x^2+x√2+1)(x^2-x√2+1)
1 1
=(1/2)(-------------- + ----------- )
x^2+x√2+1 x^2-x√2+1
1
∫--------------dx
x^2+x√2+1
1
=∫-----------------dx
(x+1/√2)^2+(1/2)
√2
=∫-----------------d(x√2+1)
(x√2+1)^2 + 1
= √2arctan(x√2+1)
同理
1
∫--------------dx
x^2-x√2+1
=√2arctan(x√2-1)
所以積分的結果為
1
=[(√2)/2][arctan(x√2+1)+arctan(x√2-1)]│
0
=[(√2)/2][(3/8)π-π/4+π/8+π/4]
= (π√2)/4
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 122.124.106.150
推
07/09 00:47, , 1F
07/09 00:47, 1F
推
07/09 00:53, , 2F
07/09 00:53, 2F
→
07/09 00:57, , 3F
07/09 00:57, 3F
→
07/09 00:58, , 4F
07/09 00:58, 4F
推
07/09 11:02, , 5F
07/09 11:02, 5F
→
07/09 11:12, , 6F
07/09 11:12, 6F
→
07/09 11:12, , 7F
07/09 11:12, 7F
討論串 (同標題文章)