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# FP3 Hyperbolic Integration. watch

1. The problem is i'm having difficulty proving the following identity:

If that's even right.

I kind of get stuck at

Thanks

Sohan
2. Try expanding .
3. Why dont you say cosh^2(x) = 1 + sinh^2(x)

cosh^3(x) = cosh(x) + cosh(x)sinh^2(x)

Then cosh(x) integrates to give sinh(x)
and cosh(x)sinh^2(x)... well use the substitution t = sinh(x)

And then you don't even need to bother proving that identity
4. (Original post by Glutamic Acid)
Try expanding .
Yeah but that's practically cheating. In the exam, I won't have the answer to work backwards from.
5. Just see . Then first term is easier, and second term is a function to a power, next to its derivative (with some coefficient)
6. (Original post by tazarooni89)
Why dont you say cosh^2(x) = 1 + sinh^2(x)

cosh^3(x) = cosh(x) + cosh(x)sinh^2(x)

Then cosh(x) integrates to give sinh(x)
and cosh(x)sinh^2(x)... well use the substitution t = sinh(x)

And then you don't even need to bother proving that identity
Yeah I neevr thought to do it like that

Thanks
7. (Original post by sohanshah)
Yeah but that's practically cheating. In the exam, I won't have the answer to work backwards from.
Oh, I see, I thought that result was in the question. Use Kolya's method.
8. Any idea as to how to
9. Well... first you need to know how to integrate cosech(x) and then it's easy

If you treat cosech(x) as a fraction, [cosech(x)]/1

then multiply the top and bottom of that fraction by coth(x) - cosech(x), you should notice that the top of the fraction is the differential of the bottom.

Does that help?
10. Use the t-substitution. If you aren't sure how to formulate it for hyperbolic, just write , take the factor of i outside of the integral sign, integrate using the t-substitution, and then turn it back into hyperbolic functions afterwards.
11. Multiply top and bottom by
12. (Original post by tazarooni89)
then multiply the top and bottom of that fraction by coth(x) - cosech(x), you should notice that the top of the fraction is the differential of the bottom.
Divine inspiration, eh?
13. Half tanh substitution:

14. (Original post by Kolya)
Divine inspiration, eh?
lol, no that's just how my FP3 teacher taught it

If I were integrating cosech(x) during my university course, I'd probably use the substitution t = sinh x, which makes it work out quite nicely, although I don't think an FP3 question requires you to be super-rigorous about it.
15. (Original post by Mr M)
Half tanh substitution:

How did you get from step 2 to 3??

I'm basically self-teaching this so don't think i'm stupid. Even though I am!
16. (Original post by sohanshah)
How did you get from step 2 to 3??

I'm basically self-teaching this so don't think i'm stupid. Even though I am!

From the half tangent subs, you just apply Osbourne's rule (so the signs change for ).

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