First-year calculus questions

[albema]

I have 2 questions to ask.

Which is “integrand” and the “antiderivative” on the following integral?
\[\int dx / {1 + (tan x)}\]

What is “hyperbolicus” (form of trigonometry identities which include e or exponential)?

I am an undergraduate student of engineering, so please tell me the easiest simplest way on this. I see many people here are advance.

Thank you

 

[Dmitrij]

hyperbolicus is the the class of special functions, which are defined the following way, as it's written below:

sh(x) = 0.5 * (exp(x)-exp(-x))

ch(x)= 0.5 * (exp(x)+$exp(-x))

th(x) = sh(x)/ch(x)

cth(x) = ch(x)/sh(x)

There are also several formulas which establish the links between various hyperbolic functions. For example, it's possible to express the hyperbolic function of the double argument through the ones of the unit argument. As in trigonometry, there is also the main hyperbolic identity:

ch(x)^2 - sh(x)^2 = 1

Concerning the information sources on this subject, you can look in Matlab's electronic documentation. Here you'll manage to find the description of these functions, their properties and also easily plot their graphics with the corresponding Matlab commands. It'll be very useful and convenient for reminding.

With respect,

Dmitrij

 

[Kral]

albema,
The integrand is 1/(1+tan(x)).
.
The antiderivative (indefinite integral) is
x/2+[ln(|sinx +cosx|)]/2 + C
.
Where C is an arbitrary constant.