# Integration Bee

1. Mar 18, 2005

### phoenixthoth

And, yes, I mean what you think it means: antidifferentiating. Finding a primative. Whatever...

At my school, we're conducting an integration bee not unlike similar bees done elsewhere.

The purpose of this thread is two-fold:
1. To compile a list of non-routine integrals
2. To discuss how these integrals are done

Here is the list I came up with. Please add to the list. I'm not looking for 300 integrals no one at a community college can solve (and, yes, by "solve", I mean to antidifferentiate). They should stump a significant percentage of those who would get an A in Integral Calculus though not 100%.

A few of these are downright easy but they can stump the woefully inexperienced.

One or two of them are potentially very difficult if you're not clever enough.

2. Mar 18, 2005

### marlon

What is the most difficult one according to you ?

i will solve it

marlon

3. Mar 18, 2005

### mruncleramos

Most are standard textbooks problems.

4. Mar 18, 2005

### phoenixthoth

I would guess that #9 would stump the most people but I really have no idea.

You may find #10 and #2 fun as well.

5. Mar 18, 2005

### Crosson

Those are nice, but the tricks don't really get fun until definite integration:

$$\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}$$

$$\int_{-\infty}^{\infty}\frac{sin(x)}{x} dx = \pi$$

The first one is a fairly trivial change of variable trick, and the second is found by contour integration. In addition to having these down pat, an integrator champ should be able to pull all the tricks: differentiating under the integral sign, direct integration of differential eq corresponding to the function, etc.

Last edited: Mar 18, 2005
6. Mar 18, 2005

### marlon

9 is probably the most easy one

2 is easy with a good substitution and ten is nice

marlon

7. Mar 18, 2005

### Tom Mattson

Staff Emeritus
I like these:

$$\int e^{ax} sin(bx)dx$$

$$\int e^{ax} cos(bx)dx$$

$$\int sec^n(x) dx$$

(n is odd and positive in the last one).

They all have one thing in common: when integrating by parts, you have to recognize that you eventually get the same integral you started with, and you have to add a multiple of it to both sides to finish the problem.

Last edited: Apr 1, 2005
8. Mar 18, 2005

### phoenixthoth

You think 9 is easier than 4, 5, 14, 15, and 19? Hmm... Is there an easy way to do #9 that I don't know about? What's the easy way to solve #9?

9. Mar 18, 2005

### marlon

9 is equal to

$$\int \frac{dx}{(x^2 + 8)^2 -16 x^2}$$

then use a²-b² = (a-b)(a+b) in the denominator
then integration by partial fractions

marlon

10. Mar 18, 2005

### marlon

9 is easy because it is quite straightforeward this is the easiest way out

marlon, though i admit it requires some calculation

11. Mar 18, 2005

### phoenixthoth

Yeah that's how I'd do it but I wouldn't call factoring x^4+64 "easy." Sure, compared to the proof of FLT it's easy but... ;)

12. Mar 18, 2005

### marlon

how did you do 10 ?

marlon

13. Mar 18, 2005

### marlon

i got an answer by using the substitution t²=tanx and then apply partial fractions

marlon

14. Mar 18, 2005

### phoenixthoth

I'm no integral bee champion so this is probably not the best way to do it:

u=Tan[x] turns it into
Sqrt/(1+u^2).

Re-write:
u/(Sqrt(1+u^2)).

Now let v=Sqrt to get
2v^2 / (1+v^4) which is integrable by partial fractions as the denominator splits into (v^2+Sqrt[2]v+1)(v^2-Sqrt[2]v+1).

15. Mar 18, 2005

### marlon

any other questions ?

marlon

16. Mar 21, 2005

### dextercioby

$$\int e^{tan x} \ dx =... ?$$

$$\int \sqrt{\sin x} \ dx =... ?$$ Piece of cake for champions like Marlon

17. Mar 21, 2005

### marlon

18. Mar 21, 2005

### marlon

The integral exp(tanx) is not that difficult, if i am right. Write the exp(tanx) as a series :
sum over n of ((tanx)^n)/n!

Now the integral sign and summation sign can be interchanged so what we really need to integrate is (tan(x))^n...

i suppose that by writing tan as sin/cos, you can construct a recursion relation. This relation can be constructed by integration by parts...

marlon

19. Mar 21, 2005

### motai

holy... geez talk about scaring a Calc 1 student... I don't even know how to approach some of those integrals... geez. The only thing that comes close that I can solve is probably applying an arctangent rule to no. 9 from my knowledge, but as marlon said I probably have to review partial fractions.

ack.

20. Mar 21, 2005

### dextercioby

Marlon,the $\int e^{\tan x} \ dx$ admits an exact (nonseries) solution among special functions.

Daniel.