TRIGONOMETRIC PRIMITIVES

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TRIGONOMETRIC PRIMITIVES :

TRIGONOMETRIC PRIMITIVES by A ndreas – W alter Zimmerman Centrul de Excelen ţă pentru Tinerii Capabili de Performan ţă Boto ş ani Good Afternoon and welcome to our school! I am going to present you an interesting problem, which I had to do during a contest last week. It is from my favourite maths chapter, mathematical analysis and I hope you will enjoy my solution.

And here is the problem: :

And here is the problem: Find the primitive: For x (- , ).

First observations.:

First observations. Firstly, we can see that the definition domain of our function is (- , ), the ends of the interval being the points in which the function tangent reaches the values -1, respectively 1. Secondly, the numerator is = , the derivative of the expression in brackets from the denominator. This leads us to the method of integration by parts, which is the one I used in order to solve the problem.

Processing the expression:

Processing the expression = = = =

Continuing…:

Continuing… = = = = = .

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We can easily observe that the last term of the expression is exactly the primitive we have to find. Let’s denote it by I. So we obtain the equation: I= -2I=

The last step…:

The last step… We have attained an easier form of our primitive, and now we have to take the last step: finding the primitive . = = .

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Thus, we know that = = , so we can choose a function u: (- , ) , with u (x)= 1+tan x. The last primitive becomes: = .

Final replacements:

Final replacements By replacing in the equation 1, we get: -2I= = = = => I= , which is the primitive we were searching for.

The end:

The end I hope you enjoyed my solution to this pretty interesting problem! Thank you! Andreas Zimmermann

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