Techniques of Integration: Trigonometric substitutions

The familiar trigonometric identities

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may be used to eliminate radicals from integrals. Specially when these integrals involve tex2html_wrap_inline29 and tex2html_wrap_inline31 .

1
For tex2html_wrap_inline33 set tex2html_wrap_inline35 . In this case we talk about sine-substitution.
2
For tex2html_wrap_inline37 set tex2html_wrap_inline39 . In this case we talk about tangent-substitution.
3
For tex2html_wrap_inline41 set tex2html_wrap_inline43 . In this case we talk about secant-substitution.

The expressions tex2html_wrap_inline45 and tex2html_wrap_inline47 should be seen as a constant plus-minus a square of a function. In this case, x represents a function and a a constant. For example tex2html_wrap_inline53 can be seen as one of the two previous expressions. Indeed, if we complete the square we get

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where tex2html_wrap_inline57 . So from the above substitutions, we will set tex2html_wrap_inline59 .

The following examples illustrate how to use trigonometric substitutions :

[Calculus] [Next Example]
[Geometry] [Algebra] [Trigonometry ]
[Differential Equations] [Complex Variables] [Matrix Algebra]

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Author: Mohamed Amine Khamsi

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