Trigonometric Substitutions: Example 2

Evaluate

displaymath33

Solution. First let us complete the square for tex2html_wrap_inline35 . We get

displaymath37

which suggests the secant-substitution tex2html_wrap_inline39 . Hence we have tex2html_wrap_inline41 and tex2html_wrap_inline43 . Note that for x=0, we have tex2html_wrap_inline47 which gives t=0 and for x=3, we have tex2html_wrap_inline53 which gives tex2html_wrap_inline55 . Therefore, we have

displaymath57

Using the trigonometric identities (you will find them at the end of this page), we get

displaymath59

The technique of integration related to the powers of the secant-function gives tex2html_wrap_inline61 and

displaymath63

which implies

displaymath65

One would check easily that

displaymath67


Useful trigonometric identities:

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Author: Mohamed Amine Khamsi

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