TRIGONOMETRIC SUBSTITUTION

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Presentation transcript:

TRIGONOMETRIC SUBSTITUTION

TRIGONOMETRIC SUBSTITUTION NOTE: We convert the radical into a cos function NOTE: We remove the radical

TRIGONOMETRIC SUBSTITUTION

TRIGONOMETRIC SUBSTITUTION In this section, we consider integrals containing radicals Example Find The main idea is how to get rid of the radicals. ( use trig. Substitution)

TRIGONOMETRIC SUBSTITUTION

TRIGONOMETRIC SUBSTITUTION triangle domain Identity substitution Expression

TRIGONOMETRIC SUBSTITUTION Final061

TRIGONOMETRIC SUBSTITUTION

TRIGONOMETRIC SUBSTITUTION Final152

TRIGONOMETRIC SUBSTITUTION Expression

TRIGONOMETRIC SUBSTITUTION We can transform the integrand into a function for which trigonometric substitution is appropriate by first completing the square under the root sign substitution Expression

TRIGONOMETRIC SUBSTITUTION

TRIGONOMETRIC SUBSTITUTION In the following table we list trigonometric substitutions that are effective for the given radical expressions because of the specified trigonometric identities.

TRIGONOMETRIC SUBSTITUTION

TRIGONOMETRIC SUBSTITUTION

TRIGONOMETRIC SUBSTITUTION Example substitution Expression Find

TRIGONOMETRIC SUBSTITUTION EXAM-2 Term-092

EXAM-2Term-102 substitution Expression

TRIGONOMETRIC SUBSTITUTION Expression Example Find

Find TRIGONOMETRIC SUBSTITUTION hyperbolic substitutions Example can be used in place of trigonometric substitutions and sometimes they lead to simpler answers. But we usually use trigonometric substitutions because trigonometric identities are more familiar than hyperbolic identities.

TRIGONOMETRIC SUBSTITUTION Because the ellipse is symmetric with respect to both axes, the total area is four times the area in the first quadrant

substitution Expression

substitution Expression

TRIGONOMETRIC SUBSTITUTION

TRIGONOMETRIC SUBSTITUTION