Integrating by substitution

Hi, can someone confirm that what I’m doing is correct.
Would I then proceed to integrate by parts on
u x cos^2u

Looks like you have a typo going from line 2 to line 3 and not sure why youd then do line 4 (you seem to reverse this later on).

Thanks for spotting that!
Is this better
and then do u x sec^2(u) by parts?

Sure, seems reasonably straightforward how to do it.

Great thanks
Also any ideas on integrating (lnx)^2

Presming youve done the integral of ln(x)? If so, a similar approach should work?

I tried substitution but I think it’s longwinded and got to a point where I couldn’t simplify.
By parts worked which is confusing since I’ve been told to only use by parts when you have functions from different families

The usual approach to integrating ln(x) is by parts and Id suggest the same here (with the same "trick").

Wow! Worked a treat, thanks
Generally, how do I recognise what would be the best way to integrate something that looks complicated

There are no perfect rules, but if you have to integrate some functions of ln(x) so something like sin(ln(x)) or (ln(x))^n, so a function of ln(x), then when you differentiate it using the chain rule, the usual by parts trick of 1*sin(ln(x)) will "work"

But if the integrand is a product of two functions, then it could be that either by parts or a substitution (reverse chain rule) would work and you simply have to try/think about it.

gotcha thanks
does anything immediately jump out at you for the integral:
(9 - 25x^2)^3/2
Nothing I’ve tried works but it’s possible by substitution

Is that ^(3/2) or ^3 and all divided by 2? If its the former, it looks about it a bit like
(1-x^2)^(1/2)
so can that be adapted?

gotcha thanks
does anything immediately jump out at you for the integral:
(9 - 25x^2)^3/2
Nothing I’ve tried works but it’s possible by substitution

^(3/2)
Not sure what you’ve done with the adaptation

I would try integral by substitution with u=asin(5x/3)...

Why does this substitution work?

x = 3/5 sin(u)
so what happens when you sub it in? That should explain why it works?

It does work but it’s a bit bizarre that this it’s come out of seemingly nowhere

Why does this substitution work?