Question

How would you approach this integral?

Answer 1

\(\Huge \int \frac{x}{\sqrt{x^4-1}} dx \)

Answer 2

I wish I could help :O

Answer 3

Here's my thinking, step by step: I'd start by looking for substitutions.
1. Try u = x^4. Tried this substitution but it didn't help much.
2. Try u = x^2. This gives integral of 0.5*(u^2-1)^-0.5. Helpful. Let's take this forward.
3. Notice the u^2 - 1 on the bottom. If I substitute u = cosh(theta), then u^2-1 becomes sinh^2 (theta). Furthermore, the u*du on the top becomes sinh(theta)d(theta), which cancels a bit. Nice!
4. Let me work the next bit out on paper, then I'll get back to you... :)

Answer 4

Oops, I missed a square root. In fact, step 3 gives sinh(u)/sinh(u), which is just 1. So step four is just integrate 1 w.r.t. theta, which gives theta.

Answer 5

Now unravelling the substitutions in reverse order, u = x^2 and u = cosh(theta). So theta becomes arccosh(u) which is arccosh(x^2). So the final answer is arccosh(x^2), I think!

Answer 6

(I can write this up properly with the maths/equation/LaTeX notation, if you want)

Answer 7

Thanks for your input, it was arccosh (x^2) +C actually :D

I see what I did wrong tho.. I used a bad substitution :/

Answer 8

Oops, yes, I always forget the +C !
Substitutions are a pain, it's often a case of trial and error. Apparently it gets better with "experience", but I reckon that probably means doing enough practice problems to remember most of them :/

Answer 9

True, thanks for your help, really appreciate it (:

Answer 10

No problem! :)