The Method of Trig Substitution

The Method of Trig Substitution - The Method of...

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Unformatted text preview: The Method of Trigonometric Substitution Main Idea The method helps dealing with integrals, where the integrand contains one of the following expressions: (where a and b are constants) b ax bx a bx a- +- 2 2 2 A simpler forms of the former expressions are the following ones : 1 1 1 2 2 2- +- x x x To get rid of the root, we substitute sin , tan or sec respectively The radical Substitution The radical becomes dx becomes x = sin cos cos d x = tan sec sec 2 d x = sec tan sec tan d 2 1 x- 2 1 x + 1 2- x To apply that to the general cases, we transfer the radical to a form similar to the respective simple form 2 2 2 2 ) ( 1 x a x b a a b- =- 2 2 2 2 ) ( 1 x a x b a a b + = + 1 ) ( 2 2 2 2- =- x a a x b a b The First Case We let: sin = (b/a)x x = asin /b dx = (a/b) cos d The radical becomes a cos 2 2 2 2 ) ( 1 x a x b a a b- =- The Second Case We let: tan = (b/a)x x = (a /b) tan dx = (a/b) sec 2 d The radical becomes a sec 2 2 2 2 ) ( 1 x a x b a a b + = + The Third Case We let: sec = (b/a) x x = (a /b) sec dx = (a /b) sec tan d The radical becomes a tan 1 ) ( 2 2 2 2- =- x a a x b a b Examples...
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The Method of Trig Substitution - The Method of...

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