7.3 Trigonometric Substitution

7.3 Trigonometric Substitution - 7.3 Trigonometric...

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7.3 Trigonometric Substitution In this section, we use the method of trigonometric substitution to evaluate integrals involving the radicals , 2 2 x a - , 2 2 x a + 2 2 a x - and where a > 0 REFER to the handout for Trigonometric Substitution The objective with trigonometric substitution is to eliminate the radical in the integrand by using the Pythagorean identities.
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Examples: Evaluate the integral. 1) 2) - 9 16 2 2 x x dx dx x x - 2 2 5 dx x x + 2 0 2 3 4 3) ( 29 - 2 / 3 2 4 x x dx 4) 5) + - 13 6 2 t t dt
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- 9 16 2 2 x x dx let θ θ θ θ π θ π π θ θ d dx x x tan sec 4 3 sec 4 3 2 3 or 2 0 , sec 3 4 = = < < = θ θ θ θ tan 3 tan 3 tan 9 9 sec 9 9 16 2 2 2 = = = - = - x and Note: 0 tan θ because 2 3 or 2 0 π θ π π θ < < ( 29 + = = = = - C d d d x x dx θ θ θ θ θ θ θ θ θ θ sin 9 4 cos 9 4 sec 1 9 4 tan 3 sec 16 9 tan sec 4 3 9 16 2 2 2 3 θ 4 x 9 16 2 - x Use the right triangle to covert back to x ’s θ C x x C x x + - = + - = 9 9 16 4 9 16 9 4 2 2
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