# Really hard integrals?

1. Jun 11, 2007

### pakmingki

can someone give me some really hard intergrals to solve?

make sure they are in the range of calculus 1-2 (anything before multivariable)

My teacher assigned some few hard integrals, and they are fun. I want to try moer.
thanks.

2. Jun 11, 2007

### yip

Try $$\int{\frac{(1+x^{2})dx}{(1-x^{2})\sqrt{1+x^{4}}}}$$

Last edited: Jun 11, 2007
3. Jun 11, 2007

### itsjustme

sin(2x)cos(2x)dx

4. Jun 11, 2007

### ObsessiveMathsFreak

$$\int e^{-x^2} dx$$

5. Jun 11, 2007

### VietDao29

I doubt that it belongs to either Calculus 1 or Calculus 2 problems.

6. Jun 11, 2007

### Gib Z

$$\int^{1}_0 \frac{\log_e (1+x)}{x} dx$$. Quite an interesting one that someone gave to me. Nice Solution :)

7. Jun 11, 2007

### Kurdt

Staff Emeritus
Find $$\frac{f'(x)}{f(x)}$$ where $$f(x) = sec(x)+tan(x)$$and hence find $$\int sec(x) dx$$.

One of my faves

8. Jun 11, 2007

### pakmingki

wow, these loko pretty fun. THey look way different from the ones ive ever seen.

Ill give them a whirl sometime soon.

9. Jun 11, 2007

### bob1182006

This is a pretty hard one but I haven't finished Calc 2 so I don't know any harder than this.

My favorite Integral so far is this:

$$\int \frac{dx}{(x^2+9)^3}$$

It's general form is of
$$\int \frac{dx}{(x^2+a^2)^n}$$

It has a really interesting answer

10. Jun 12, 2007

### zoki85

Hard ,but famous and bautiful :

$$\int_{0}^{\infty}sin(x^2)dx$$

11. Jun 13, 2007

### janhaa

Try this one...

$$\int \sqrt{\tan(x)}{\rm dx}$$

12. Jun 13, 2007

### bit188

That's a good one :rofl:

13. Jun 14, 2007

### Invictious

Took me 5 minutes only
That question though, however, was just..simply amazing.
I suggest everyone try that question

14. Jun 14, 2007

### Gib Z

I think the original poster has quite enough thanks...he hasn't actually done any of them yet.

15. Jun 14, 2007

### JohnDuck

I'm stumped but intrigued.

16. Jun 14, 2007

### Equilibrium

$$\int \frac{1}{x^5+1}dx$$

17. Jun 14, 2007

### zoki85

We are not all as clever as you Invictious :tongue:

18. Jun 14, 2007

### DyslexicHobo

I'm just out of Calc 1, so I'm not sure if I even have the knowledge to solve this... but here's where I am now. I can't tell if I complicated it even more, or if I'm closer to getting the solution.

$$\int (r-1)^{1/5}ln(r) - r^{-1} (r-1)^{1/5} dr$$

I used u-substitution (well, r-substitution), where $$r = x^5 + 1$$. After the substitution I used integration by parts, and now I'm unsure if that was even the right path. Please let me know!

And please tell us how to do $$\int_{0}^{\infty}sin(x^2)dx$$

Last edited: Jun 14, 2007
19. Jun 14, 2007

### prasannapakkiam

how can this even be integrated?:uhh:

20. Jun 14, 2007

### JohnDuck

It can be proved that there's no elementary antiderivative, but you can use a trick from multivariable calculus involving a change to polar coordinates and the squeeze theorem to evaluate it. It's called a Gaussian integral.

Edit: Correction--the trick works for $$\int_{- \infty}^{\infty} e^{-x^{2}}dx$$

Last edited: Jun 14, 2007