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# Calc08ex - Ch.8 Extra Topic Trigonometric Substitutions...

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Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2004 Ch.8 Extra Topic: Trigonometric Substitutions Monticello (Thomas Jefferson’s home), Alexandria, VA

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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 + = 2 2sec 4 x θ= + tan 2 x 2 tan x 2 2sec d dx 2 2sec 2sec d sec d ln sec tan C + + 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 sec d ln sec tan C + + 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 + + +

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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 -
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Calc08ex - Ch.8 Extra Topic Trigonometric Substitutions...

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