Can you skip trig substitution?

Tags:
1. Sep 5, 2015

Ramanujan143

Hi, I'm currently taking ap calc bc as a senior in high school. Since trig sub and power reduction formula is not apart of the ap curriculum our class is skipping it. Assuming I pass the test and get credit for it, I will probably skip calc 2 in college. If I continue to study math and physics in college will the fact that I don't know trig sub put me at a disadvantage; will I need to learn trig sub in college? Or can i just never learn it? Thank you.

2. Sep 5, 2015

SteamKing

Staff Emeritus
The thing about trig substitution is, you never need it until you do.

If you are the calculus whiz you think you are, you should be able to master it.

3. Sep 5, 2015

Ramanujan143

I hope so. Thank you. Also, I don't think I was implying that I am a calculus whiz ha-ha; I don't think I am.

4. Sep 6, 2015

RJLiberator

I learned trig sub in calculus 2. I have yet to use it once since. Although I am not 'that' advanced yet, it does seem to be something that is un-needed.

With that being said, you will, personally, want to know trig substitution and have done it a few times. It's just interesting and broadens your mathematical understanding of integration.

5. Sep 6, 2015

Staff: Mentor

I disagree. Trig substitution is extremely useful any time the integrand contains the square root of a sum or difference of squares. The radical can be in either the numerator or denominator.

6. Sep 6, 2015

Hornbein

Go to Boston Common and look at the winos. None of them learned trig substitution. See where they are now.

7. Sep 6, 2015

RJLiberator

I speak from academic progress not pursuit of knowledge/intelligence.

Trig substitution is very worthy of learning for the pursuit of knowledge.

But after I learned it in calculus 2, it has not shown up in any of my physics 1 or 2 or 3 courses, calc 3, linear algebra, proof based math courses, complex analysis, etc.

8. Sep 7, 2015

Ramanujan143

Thanks for all the replies guys. I have done a few problems of trig sub, but Ican't say that I fully learned it. I remember doing something very similar in physics when integrating the biot savart law\equation. I think I will skip it for now, but if I ever feel I need it, I'll learn it on the fly.

9. Sep 7, 2015

micromass

Staff Emeritus
Your courses must be very bad then, because it shows up ALL the time.

10. Sep 7, 2015

Staff: Mentor

Trig substitution probably wouldn't show up in physics 1 at all, and very little or none in the other physics courses. You don't find much calculus in linear algebra, other than possibly some differentiation, nor would there be much calculus in a course on proofs. Of the courses you listed only the calc 3 course and the complex analysis would be likely candidates for problems that might require trig substitution. This technique would probably show up in a course on differential equations, which you didn't list.

Since you don't feel that competent at trig substitution, I would advise taking another look at some problems now, rather than later.

11. Sep 7, 2015

mathwonk

suppose you want to compute a fairly basic example, the area of a circle, x^2 + y^2 = r^2, or y = sqrt(r^2-x^2). the integral ydx can be computed by setting x = rsin(t). and dx = rcos(t)dt, giving an integral of r^2.cos^2(t)dt.

then using parts, cos^2(t)dt = cos(t).dsin(t) so the integral is sin(t)cos(t) - integral of
sin(t).dcos(t) = sin(t)cos(t) + integral of sin^2 = sin(t)cos(t) + integral of (1 - cos^2).

So 2.integral (cos^2) = sin(t)cos(t) + t. and since t = arcsin(x/r), this is

2.integral (cos^2(x)) = (x/r).sqrt(1- x^2/r^2) + arcsin(x/r). so integrating from say x=0 to x=r, gives arcsin(1) = π/4, and multiplying by r^2 gives for the area (of one quarter of a circle) as: π.r^2/4. How would you do that without a trig substitution?

There really isn’t much to learn here. Then only point is that the trig identities: 1+tan^2 = sec^2 and cos^2 = 1- sin^2, let you change any square root like sqrt(1+x^2) or sqrt(1-x^2), into a perfect square under the radical by replacing x by sin(t) or tan(t). I.e. 1+x^2 is not a perfect square, but 1+tan^2 is. You can handle more complicated degree two expressions by the usual high school algebra of completing the square. So this technique is also good practice in a fundamental algebraic tool that is useful in zillions of places, namely completing the square. So you really should learn it.

12. Sep 14, 2015

CrystalCaribean

<-- Late to the party.

I'm currently taking Calc II and we JUST finished the section on trig substitutions last week. They can be fairly handy when integrating, but you could definitely teach yourself how to do them, so long as you have a working knowledge of trigonometry. If you remember all of your sin/cos/tan identities, derivatives, and integrals (our entire class had to remove the cob webs/dust from that portion of our brains almost instantaneously), you can easily watch a few youtube videos and see how substituting makes certain integrals much easier to solve.

Skipping it entirely? I'm not sure I'd go that route. It's always nice to have all the tools you need to solve the problems that come your way :)

13. Sep 15, 2015

mpresic

In my Physics experience of over 40 years, Trig substitution is one of the most common and important techniques you will use, If integration by parts is what is referred as power reduction, this is important too. It is a lot better to learn these techniques now (in secondary school). In college, you will be much busier if you are in a science program. The competition will be tougher, and will be familiar with these techniques. Sounds like CrystalCaribean is right on. When you come with powerful mathematical tools, learning physics is easier, not harder to learn,

14. Sep 16, 2015

MostlyHarmless

Like someone said, you never need it until you do. Once you get into DiffEq it will be very useful.

Also, if there is a problem in a textbook that needs it, (there will be). Those problems are nearly impossible to do with out it.

That being said, it's really not hard at all. Memorize your Pythagorean identities and you're golden.

15. Sep 17, 2015

CrystalCaribean

As I'm beginning to figure out this semester (3 hours into tonight's homework assignment), it isn't the calculation or the remembering of identities that's hard. The "hard" part is staring at an integral and knowing which integration method is the best one to use and WHY that method is the best choice, unless, of course, you REALLY enjoy taking 90 minutes to solve each integral.

My math experience is quite insignificant compared to some of the others replying here, but it seems that Calc II forces you to become much more concerned with the "why" than the "how" and I've found that a bit more challenging.

Also noteworthy, there are two younger boys in my Calc II class who just graduated high school. This is their first college math class, and there have already been a few instances in which they missed out on something in their high school Calc I class that they now need in this course. I'm not sure where you live, but here in southern California, it seems to be an issue. Might be worth looking into so you have a better handle on which blanks you'll need to fill?