微積分高手幫忙一下

1.

(1) Sdx/[(x-1)(x^2-2x-3)^1/2]dx

=S1/{(x-1)[(x-1)^2-4]^1/2}dx [令x-1=2sect 故dx=2tant*sect dt 代入]

=S{1/{2sect*2tant]}*2tantsectdt

=1/2Sdt

=1/2 t+c [因x-1=2sect, sect=(x-1)/2 t=sec^-1[(x-1)/2]代入]

=1/2sec^-1(x-1)/2+c
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2. Sx/根號2-x-x^2 dx

=Sx/[2-x-x^2]^1/2dx

=Sx/[-[(x+1/2)^2-9/4]^1/2dx [令x+1/2=3/2sect 則dx=3/2tant sectdt代入]

=-S[(3/2sect-1/2)/(3/2*tant)]*3/2*tantsectdt

=-S[3/2*sec^2t-1/2sect]dt

=S1/2sect-3/2*sec^2tdt

=1/2*lnlsect+tantl-3/2*tant+c

[x+1/2=3/2sect sect=2/3*(x+1/2) tant=[4/9*(x+1/2)^2-1]^1/2

=2/3*(x^2+x-2)^1/2代入]

=1/2*lnl{2/3*(x+1/2)+2/3*(x^2+x-2)^1/2}l-(x^2+x-2)^1/2+c

第1題不是已有人解了?

第2題你前一問說 x 要改成 dx,
如果還是 x/√(2-x-x^2) 的積分,
x/√(2-x-x^2) = (1/2 + x)/√(2-x-x^2) -(1/2)/√(2-x-x^2)
第一項簡單代換: u = 2-x-x^2,
第二項代換法同前一問.