Trigonometric Substitutions: Example 1

Find

displaymath34

Solution. It is easy to see that sine-substitution is the one to use. Set tex2html_wrap_inline36 or equivalently tex2html_wrap_inline38 . Then tex2html_wrap_inline40 which gives us

displaymath42

Easy calculations give

displaymath44

Technique of integration of powers of trigonometric functions give

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which suggests the substitution tex2html_wrap_inline48 . Hence tex2html_wrap_inline50 which implies

displaymath52

Therefore, we have

displaymath54

This will not answer fully the problem because the answer should be given as a function of x. Since tex2html_wrap_inline58 , we get after easy simplifications

displaymath60

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