Tuesday, June 21, 2011

6.4 – Derivatives & Integrals of Hyperbolic & Inverse Hyperbolic Functions

The derivatives and integrals of hyperbolic functions and inverse hyperbolic functions are very similar to those of trigonometric and inverse trigonometric functions, just with a difference of a negative sign somewhere within the formulas. There is no rule that we can tell where the minus sign has changed, so this section requires a lot of memory work.


HYPERBOLIC FUNCTIONS

The derivatives of hyperbolic functions can be derived easily by converting the functions into their exponential form. I’ll leave it for you as an exercise to derive all of them. The list of derivatives are as follows:
image

As you can see, the derivative of sinh x is cosh x, and vice versa, which is different from trigonometric ones by a minus sign. The functions whose derivatives have minus signs are the secondary hyperbolic functions, csch x, sech x and coth x.

The integrals, again, are very similar to trigonometric integration.
image

The integrals for sech x and csch x may look a little weird. You should try to differentiate the right hand side and see whether you get the expression on the left. Again, you should do some homework to derive all of them.


INVERSE HYPERBOLIC

Again, the inverse hyperbolic functions have similar derivatives to what the trigonometric functions have, and it is just a matter of a minus sign, with or within the square roots. Deriving is similar: derive them implicitly and make use of the hyperbolic identities (do not confuse with the trigonometric ones. Remember Osborne’s rule). Here you go
image

The integrals, as usual, are harder to do. You need to use integration by parts, as I said in the previous section. Try doing them as how you did for the previous section. As a matter of fact, the huge ‘ln’ terms in the integrals of csch-1 x and sech-1 x are just logarithmic forms of cosh-1 x and sinh-1 x.
image


This section can only be mastered by doing an adequate amount of exercises. Frankly, the integrals of inverse functions don’t need to be memorized, but you must make sure you can derive them on the spot. You may start to confuse with so many kinds of derivatives and anti-derivatives. But that’s not the end yet, as I haven’t combine some results that can be obtained from both these sections, you will see it only in the next section. Beware, the next section is not as easy…

No comments:

Post a Comment