Separating 11−x2 into multiple termsI'm working through an example that contains the following steps:∫11−x2dx=12∫11+x−11−xdx=12ln⁡1+x1−xI don't understand why the separation works. If I attempt to re-combine the terms, I get this:11+x11−x=1−x1−x11+x−1+x1+x11−x=1−x−(1+x)1−x2=−2x1−x2≠21−x2Or just try an example, and plug in x=2:211−22=−2311+2−11−2=13+1=43≠−23Why can 11−x2 be split up in this integral, when the new terms do not equal the old term?

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Separating 11−x2 into multiple termsI&#039;m working through an example that contains the following steps:∫11−x2dx=12∫11+x−11−xdx=12ln⁡1+x1−xI don&#039;t understand why the separation works. If I attempt to re-combine the terms, I get this:11+x11−x=1−x1−x11+x−1+x1+x11−x=1−x−(1+x)1−x2=−2x1−x2≠21−x2Or just try an example, and plug in x=2:211−22=−2311+2−11−2=13+1=43≠−23Why can 11−x2 be split up in this integral, when the new terms do not equal the old term?

Yaritza Phillips · Accepted Answer

The thing is11−x+11+x=21−x2What you might have seen is1x−1−1x+1=21−x2Note the denominator is reversed in the sense 1−x=−(x−1).

German Ferguson · Accepted Answer

you wrote wrong fraction :11−x2=12(11−x−11+x)instead of :11−x2=12(11+x−11−x)Now check it by plug in x=2LHS=1−3; RHS=12(13−1)⇒1−3

Separating \(\displaystyle{\frac{{{1}}}{{{1}-{x}^{{2}}}}}\) into multiple terms I'm working through

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