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Do you know what 1/sinx is? and what 1/cosx is?

since 1/sinx is the same as cscx, You should look up in your book what the integration of cscx is.

You are missing the point of the question, @zordoloom

Seconded.

I was contemplating an integration-by-parts idea after doing a substitution.

Probably, but not the mnemonic

No, looking on wikipedia I don't.

@henpen are you there? I'm waiting for a response. Whenever you're ready.

@Coolsector I thought you meant multiply the given integrand by sin x.

No, it's not v. I said it's between u and dv.

Sorry, I always think by parts as\[\int\limits u\frac{dv}{dx}dx=uv-\int\limits \frac{du}{dx}vdx\]

Oh dear, that was a stupid mistake, yes.

Sorry, I've frozen up- I don't know

You're going to be integrating that, so not particularly.

Or am I being too self serving, choosing the easiest option (technically)
?

Could you explain why the LIATE 'rule' works?

\[\int\limits \tan(x)\cos(x)x dx\]

Is the part of interest

What exactly did I do incorrectly?

What was incorrect?

Where I wrote uv the bounds were implicit.

It is the reverse of the product rule.

NO it is not!

("It is the reverse of the product rule. " True Story)

@calculusfunctions , yes it actually is.

Carry on.

@calculusfunctions , this is getting nowhere: thank you for your time.