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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 θ , tan θ or sec θ respectively The radical Substitution The radical becomes dx becomes x = sin θ cos θ cos θ d θ x = tan θ sec θ sec 2 θ d θ x = sec θ tan θ sec θ tan θ d θ 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 θ = (b/a)x → x = asin θ /b → dx = (a/b) cos θ d θ The radical becomes a cos θ 2 2 2 2 ) ( 1 x a x b a a b = The Second Case We let: tan θ = (b/a)x → x = (a /b) tan θ → dx = (a/b) sec 2 θ d θ The radical becomes a sec θ 2 2 2 2 ) ( 1 x a x b a a b + = + The Third Case We let: sec θ = (b/a) x → x = (a /b) sec θ → dx = (a /b) sec θ tan θ d θ The radical becomes a tan θ 1 ) ( 2 2 2 2 = x a a x b a b Examples...
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This note was uploaded on 10/12/2011 for the course MATH 201 taught by Professor Foad during the Spring '11 term at Qatar University.
 Spring '11
 foad
 Calculus, Integrals

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