Question

If a =
$\sqrt{10} + \sqrt{5}$
and b =
$\sqrt{10 } - \sqrt{5}$
Then find
$( \sqrt{a} - \sqrt{b} ) \div (\sqrt{10} - \sqrt{5} ) - (2 \sqrt{ab} ) \div (\sqrt{10} + \sqrt{5} )$

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Answer 1

I=∫(tanx−−−−√+cotx−−−−√)dx
I=∫(tan⁡x+cot⁡x)dx
=∫sinx+cosxsinxcosx−−−−−−−−√dx
=∫sin⁡x+cos⁡xsin⁡xcos⁡xdx
Putting sinx−cosx=u,sin⁡x−cos⁡x=u, du=(cosx+sinx)dx,u2=1−2sinxcosx,sinxcosx=u2−12du=(cos⁡x+sin⁡x)dx,u2=1−2sin⁡xcos⁡x,sin⁡xcos⁡x=u2−12

I=∫2–√du1−u2−−−−−√=2–√arcsinu+C=2–√arcsin(sinx−cosx)+C
I=∫2du1−u2=2arcsin⁡u+C=2arcsin⁡(sin⁡x−cos⁡x)+C
where CC is an arbitrary constant for indefinite integral.