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\[\int\limits 1/(1+x ^{2})^{1/2}\]
it will be \[\ln (1 + x ^{2})^{1/2}\]

but the derivative of denominator should be there in the numerator

\[x= \tan \theta\]

ok , the derivative of (x^2 )^1/2 is 1/2 (2x)
so 2 cancel the other 2 :)

and for natural log, exponent should be 1

\[\int\limits 1/x= \ln x\]

if the exponent is other than 1, we use power rule

yeah but we can't multiply it with 2x so ln(1+x²)½ is not the right answer

\[d x= \sec^2 \theta d \theta\]

i think its correct ..
anyways good luck ..

\[\int\limits \sec^2 \theta/(\sqrt{1+\tan^2 \theta} )\]

\[\int\limits \sec^2\theta/\sec \theta d \theta\]

\[\int\limits \sec \theta d \theta\]

\[\ln (\sec \theta + \tan \theta) +C\]

by backward substitution
\[\tan \theta =x\]

\[\sec \theta =\sqrt{1+x^2}\]

post it as a new Q,some one would surly look at it :)

Uzma are you from Pakistan?

ahaa ! ok waiting ur new Qs :)

yes :)

Can you please tell me your qualification and city

what would u do with that :)

nothing just to guess the level of your understanding mathematics

mathematics is as vast, that one can hardly grasp even one area