Calc08_4day2 - 8.4 day 2 Trigonometric Substitutions Greg...

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Unformatted text preview: 8.4 day 2 Trigonometric Substitutions Greg Kelly, Hanford High School, Richland, Washington We can use right triangles and the pythagorean theorem to simplify some problems. θ a x 2 2 a x + 1 2 4 dx x + ∫ These are in the same form. 2 2 4 x + 2 4 sec 2 x θ + = 2 2sec 4 x θ = + tan 2 x θ = 2tan x θ = 2 2sec d dx θ θ = 2 2sec 2sec d θ θ θ ∫ sec d θ θ ∫ ln sec tan C θ θ + + 2 4 ln 2 2 x x C + + + → We can use right triangles and the pythagorean theorem to simplify some problems. 1 2 4 dx x + ∫ 2 2sec 2sec d θ θ θ ∫ sec d θ θ ∫ ln sec tan C θ θ + + 2 4 ln 2 2 x x C + + + 2 4 ln 2 x x C + + + 2 ln 4 ln 2 x x C + +- + This is a constant. 2 ln 4 x x C + + + → a θ x 2 2 a x- This method is called Trigonometric Substitution. If the integral contains , we use the triangle at right. 2 2 a x + If we need , we move a to the hypotenuse. 2 2 a x- If we need , we move x to the hypotenuse....
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Calc08_4day2 - 8.4 day 2 Trigonometric Substitutions Greg...

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