# 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

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Original post by subbhy
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).
(edited 3 months ago)
Original post by mqb2766
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?
Original post by subbhy
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.
Original post by mqb2766
Sure, seems reasonably straightforward how to do it.

Great thanks

Also any ideas on integrating (lnx)^2
Original post by subbhy
Great thanks
Also any ideas on integrating (lnx)^2

Presming youve done the integral of ln(x)? If so, a similar approach should work?
Original post by mqb2766
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
Original post by subbhy
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").
(edited 3 months ago)
Original post by mqb2766
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
Original post by subbhy
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.
Original post by mqb2766
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
Original post by subbhy
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)
(edited 3 months ago)
Original post by mqb2766
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)

^(3/2)

Not sure what you’ve done with the adaptation
Original post by subbhy
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

I would try integral by substitution with u=asin(5x/3)...
Original post by subbhy
^(3/2)
Not sure what you’ve done with the adaptation

You should be able to scale the sin or cos that youd normally do with (1-x^2)^(1/2)? Its a standard approach for such problems.
(edited 3 months ago)
Original post by math-path
I would try integral by substitution with u=asin(5x/3)...

Why does this substitution work?
Original post by subbhy
Why does this substitution work?

x = 3/5 sin(u)
so what happens when you sub it in? That should explain why it works?
Original post by mqb2766
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
Original post by subbhy
It does work but it’s a bit bizarre that this it’s come out of seemingly nowhere

Not at all, youre doing a simpler scaling of the usual substitution for (1-x^2)^(1/2) to get to the same point. The fact that 9 and 25 are squares was a fairly strong hint, though you just need the scaling that means you can factor out a common scaling and leave it in the previous form.

A lot of harder integration problems are about thinking how you can use known standard results but apply them to slighly more complex/general cases.
(edited 3 months ago)
Original post by subbhy
Why does this substitution work?

It works because
diff(asin(5x/3),x) === 5/sqrt(9-25x^2)
(edited 3 months ago)