Further Notes on Integrating Tangents and Secants

Further Notes on Integrating Tangents and Secants - Further...

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Further Notes on Integrating Tangents and Secants In many applications, particularly those involving arclength and surface area, one needs to evaluate integrals of the general form 2 2 1 tan sec n k x xdx + , where n and k are positive integers. The technique is for accomplishing this is based on the technique of Integration by Parts and the trigonometric identity 2 2 tan sec 1 x x = - . The identity comes into play specifically as follows: ( 29 2 2 tan sec 1 n n x x = - which, when multiplied out is a polynomial in 2 sec x . Every term of the resulting integral is an odd power of the secant. Thus all these problems result in the ability to integrate odd powers of the secant. As a specific illustration of this fact, consider 4 3 tan sec x xdx : ( 29 2 4 3 2 3 tan sec sec 1 sec x xdx x xdx = - ( 29 4 2 3 sec 2sec 1 sec x x xdx = - + 7 5 3 sec 2 sec sec xdx xdx xdx = - + . The examples below illustrate how to evaluate integrals involving odd powers of the secant such as those above. It is necessary to remember that sec ln sec tan xdx x x C = + + . The last example shows how to evaluate the integral
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Further Notes on Integrating Tangents and Secants - Further...

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