How do I integrate the following: \int\frac{1+x^2}{(1-x^2)\sqrt{1+x^4}}dx

b2sonicxh 2022-01-04 Answered
How do I integrate the following:
1+x2(1x2)1+x4dx
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Expert Answer

Stuart Rountree
Answered 2022-01-05 Author has 29 answers
1+x2(1x2)1+x4dx
Let u=x1x
Then du=(1+1x2)dx
Now
1+x2(1x2)1+x4=x2(1+1x2)x(x1x)x2(x2+1x2)=1+1x2(x1x)(x1x)2+2
Thus
1+x2(1x2)1+x4dx
=duuu2+2
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Elaine Verrett
Answered 2022-01-06 Author has 41 answers
karton
Answered 2022-01-11 Author has 368 answers

Without loss of generality we may assume that 1>x>0. Put x:=y, 1>y>0. Then we obtain
1+x2(1x2)1+x4dx=1+y2(1y)1+y2ydy
Introduce the new variable
t:=1+y1y, 1<t<
Then we have
y=1+t1+t
y=2(1=t)2dt
Substituting back we obtain
1+y2(1y)1+y2ydy=t21+(1+t1+t)21+t1+t2(1+t)2dt
=12tt41dt
=122ln(t2+t41)+C
Putting back everything we obtain
122ln((1+x2)2+22x1+x4(1x2)2)+C

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