Rationalizing Trig Substitutions: Is It as Challenging as It Seems?

[suspenc3]

Okay, I am having trouble with the following, is it as hard as I think or..?There are no examples or anything in the book concerning rationalizing trig substitutions:

$$\int \frac{cosx}{sin^2x +sinx}dx$$

thanks

 

[0rthodontist]

let u = sin x

 

[Hurkyl]

Usually, there are several obvious things to try. So try them, instead of simply mulling about wondering if they'll work.

Without any ingenuity at all, you should be able to come up with

u = cos x
u = sin x
u = sin² x
u = sin² x + sin x
$u = (\cos x) / (\sin^2 x + \sin x)$

as things to try.

 

[arildno]

The substitution $u=\tan(\frac{x}{2})$ takes care of just about every contrived exercise integral you'll ever meet.

 

[suspenc3]

so let $$u=tan \frac{x}{2}$$
$$x=2tan^-^1u$$
$$dx=\frac {2du}{1+u^2}$$

and then find replacements for sinx and cosx using "u"?

 

[arildno]

That's correct!
A helpful step is to rewrite your integrand as:
$$\frac{\cos(x)}{\sin^{2}(x)+\sin(x)}=\frac{\cos^{2}(\frac{x}{2})-\sin^{2}(\frac{x}{2})}{4\sin^{2}(\frac{x}{2})\cos^{2}(\frac{x}{2})+2\sin(\frac{x}{2})\cos(\frac{x}{2})}=\frac{1-\tan^{2}(\frac{x}{2})}{4\sin^{2}(\frac{x}{2})+2\tan(\frac{x}{2})}$$
However, it might well be that some of the substitutions mentioned by Hurkyl are easier.

 

[StatusX]

u=sin(x) is the obvious and simplest choice in this case. Obvious because the integrand is clearly in the form f(sin(x))*d(sin(x))/dx. And simple because the resulting integrand is a rational function, which can be integrated by partial fractions.

If there were no other obvious substitution, then it would be a good idea to use arlidno's suggestion, because this is guaranteed to turn a rational function in any trigonometric function into a rational function in u.

 

[suspenc3]

Wait, I just tried u=sinx, and I got an answer pretty easily, is this substitution suppose to be easier?

 

[suspenc3]

yea..thanks guys

 

[suspenc3]

how about for this one:$$\int ln(x^2-x+2)dx$$

im guessing I am going to have to use partial fractions and integrateby parts. Is there any easier way to figure what I am going to substitute, I always have trouble figuring out what to sub, usually just trial and error

 

[Hurkyl]

Is there any easier way to figure what I am going to substitute,

If you practice you will eventually learn to recognize the patterns in integrands, and your intuition will help you pick out what is most likely to be useful, and try those first.

Incidentally, this integral is rather straightforward, if you remember how to manipulate logarithms.

 

[suspenc3]

i have been praticing, I still find it hard though.

 

[Hurkyl]

It changes eventually. Experience really does make a difference.

 

[prace]

The substitution $u=\tan(\frac{x}{2})$ takes care of just about every contrived exercise integral you'll ever meet.

arildno (or anyone else for that matter), can you elaborate on this comment please, or maybe point me in the direction to find an explanation myself. That seems like a very interesting concept and I think I might be able to apply that to my studies as well.

Thanks

 

[arildno]

I was exaggerating a bit; read StatusX's post for a more accurate description.

 

[suspenc3]

$$\int x cscx cotx dx$$..I simplified it a little and got

$$\int \frac{xcosx}{sin^2x}dx$$..im kinda stuck here, is it a partial fraction substitution or integration by parts, can anyone tell me if I am on the right track?

 

[arildno]

If you practice you will eventually learn to recognize the patterns in integrands, and your intuition will help you pick out what is most likely to be useful, and try those first.

Incidentally, this integral is rather straightforward, if you remember how to manipulate logarithms.

And, just adding to Hurkyl's post:
You DEVELOP that intuition; it can simply be regarded as the flashes of insight gained from condensed experience&practice, like a grandmaster in chess who might tell at a glance what might work and what won't, without doing the tedious calculations.

 

[0rthodontist]

If you practice by trying to do them mentally instead of on paper, you'll get better faster. Each problem might take you twice as long at first because you have to keep repeating the expression to yourself, but it will force you to be more aware of what you're doing.

 

[suspenc3]

yeah, I think I am going to go back and practise everyhting i learned before i go further..Ive only done like 5-10 problems in each topic...probly a good idea;)