Integration by parts (1 Viewer)

[Lukybear]

When integrating by parts, such as

integral arccosx

why cant we use let x=cos a

 

[Phylo]

bump

 

[Carrotsticks]

When integrating by parts, such as

integral arccosx

why cant we use let x=cos a

I actually don't know, but I can imagine that it has something to do with domain. I'm not sure though.



Then use IBP again? I cbb testing.

 

[Trebla]

As demonstrated by Carrotsticks, doing such a substitution leads to making the integral even more complicated.

 

[Gussy Booo]

It's a matter of practise and experience i guess.
See, if I was in your situation, I wouldn't let x=cosa anyway. I can already see it being complicated.

Sarccosx
= xarccosx - S x d(arccosx)
= xarccosx + S x/srt(1-x^2) dx
= xarccosx - 1/2 S -2x/sqrt(1-x^2)dx
= xarccosx - 1/2 S 1/sqrt(1-x^2) d(1-x^2)
= xarccosx - 1/2 (1-x^2)^(1/2) . 2 + C
= xarccosx - sqrt (1-x^2) + C

that looks much better

Thankfully, I learnt to do my integration by transformation, not substitution.

Anyways, to answer your question. It's not a matter of WHY ; its a matter of, will my integrand be EASIER to integrate through the selected substitution.

 

[fullonoob]

you'd have to integrate twice then, and when integrating by parts you dont use a substitution o-o
only u du , dv v isn't it?
that would be substitution not by parts then i guess.

 

[Drongoski]

Essentially a LaTeX version of Gussy Booo's solution.

 

[seanieg89]

If we are integrating the arc cosine function our indefinite integral need only be defined on the interval [-1,1], so the original posters substitution has no problems regarding domain and suffers from no flaw other than the fact that a smarter application of IBP yields the answer quicker.

cos u = x gives the result after two applications of IBP and then converting back to a function of x.