Miscellaneous Substitutions

Miscellaneous Substitutions - Miscellaneous Substitutions...

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Unformatted text preview: Miscellaneous Substitutions Examples I Rationalizing Substitutions dx Example (1) I= ∫ x +3 x Let x = u 6 ⇒ dx = 6u 5 du 1 2 x = (u ) = u , 6 3 3 1 3 x = (u ) = u 2 , u = 6 x 6 6u 5 du 6u 3 du ⇒I =∫ 3 =∫ 2 u +u u +1 [(u 3 + 1) −1]du = 6∫ u +1 1 2 = 6 ∫ [u − u + 1 − du u +1 u3 u2 = 6[ − + u − ln u + 1 ] + c 3 2 = 2u 3 − 3u 2 + 6u − 6 ln u + 1 + c = 2 x − 33 x − 66 x − 6 ln 6 x + 1 + c Example (2) I= ∫ dx x − 1 + ( x − 1) 3 Let x −1 = u ⇒ x −1 = u ⇒ x = u +1 dx = 2 u du 2 u du 2du ⇒I =∫ =∫ = 2 arctan u + c 3 2 u+u 1+ u = 2 arctan( x − 1) + c 2 2 Examples II Dealing with “rational functions of sinx and cosx” We use the substituti on : u = tan x 2 x ⇒ arctan u = 2 ⇒x = 2 arctan u ⇒dx = 2du 1 +u 2 1 1 = x sec 2 1 +u 2 u x 1 , sin 2 = 1 − 1+u 2 = 1 +u 2 u 1 x x sin x = 2 sin 2 cos 2 = 2 • 1 +u 2 1 +u 2 2u = 1 +u 2 1 u2 2x 2x cos x = cos 2 −sin 2 = − 2 1 +u 1 +u 2 1 −u 2 = 1 +u 2 x , cos 2 = We use the substitution : x u = tan 2 ⇒ arctan u = x 2 ⇒ x = 2 arctan u 2du ⇒ dx = 2 1+ u 1 1 , cos = = x 2 sec 2 1+ u u x 1 , sin 2 = 1 − 1+u 2 = 2 1+ u x 2 We use the substitution : sin x = 2 sin cos = 2 x 2 x 2 u 1+ u 2 • 1 1+ u 2u = 2 1+ u cos x = cos 1− u = 2 1+ u 2 2x 2 − sin 2x 2 2 1 u = − 2 2 1+ u 1+ u 2 dx Example (1) I= ∫ 3 − 5 sin x x x Let u = tan 2 ⇒ arctan u = 2 2du ⇒ x = 2 arctan u ⇒ dx = 1+ u2 2u 1− u2 sin x = & cos x = 2 1+ u 1+ u2 2u 3u 2 − 10u + 3 3 − 5 sin x = 3 − 5 ⋅ = 2 1+ u 1+ u2 2du 2du 1+ u2 I =∫ 2 =∫ 2 3u − 10u + 3 3u − 10u + 3 2 1+ u 2du I =∫ 2 3u − 10u + 3 3 1 11 = ∫ [− ⋅ +⋅ du 4 3u − 1 4 u − 3 1 1 = − ln 3u − 1 + ln u − 3 + c 4 4 1 1 x x = − ln 3 tan 2 − 1 + ln tan 2 − 3 + c 4 4 x tan 2 − 3 = ln 4 +c x 3 tan 2 − 1 dx Example (2) I= ∫ 1 + sin x + cos x 2du x Let u = tan 2 ⇒ dx = 1+ u2 2u 1− u2 sin x = & cos x = 2 1+ u 1+ u2 1 + sin x + cos x 2u 1 − u 2 2u + 2 = 1+ + = 2 2 1+ u 1+ u 1+ u2 2du 1 + u 2 = 2du = du = ln u + 1 + c I =∫ 2u + 2 ∫ 2u + 2 ∫ u + 1 1+ u2 x = ln tan 2 + 1 + c ...
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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.

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