Doubt from indefinite integration

Why aren't they following the ilate rule here?Can we take out any function Of both are easily integrable?And equally hard to integrate under the second integral sign?

ILATE is not a fixed rule .....its just for convinience some questions it is easier to solve without ILATE

It has used ilate in first part because another term generated in first integral will be cancelled by second integral

ILATE has failed in Jee Questions as well it's better to practice questions without ILATE.

Because ILATE isn’t a rule actually , rather it is a trick , that applies to most of the questions, but in some question this trick doesn’t work and instead you have to apply your skills to think that which of the function must be assumed as u/v

Thank you for clearing my doubt..

BTW that means can we take any function as first function to solve a sum easily?

You can take any function to solve it , but that won’t be easy , ILATE is used because it helps us to determine selection of which function will lead us to easier/solvable process .

See , among u,v we selected the function which is more difficult to integrate as u .
And we know , that inverse trigo function are most diff to integrate (generally), and exponential functions are most easily to integrate(generally) , so someone came out with a trick that in general we’ll consider inverse function (I) as u in most case , and we’ll assume exponential (E) as u only when no other function is left .
And so on a order ILATE was developed , based on how difficult the function is to integrate (generally )

So this works most of the times but sometimes you need to think on you own that which function should be considered u/v

I hope it’s clear now

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Yeah it's clear now...Thank you