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Miscellaneous Substitutions

Miscellaneous Substitutions - Miscellaneous Substitutions...

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Miscellaneous Substitutions
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Examples I Rationalizing Substitutions
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c x x x x c u u u u c u u u u du u u u u du u u du u u u du u I x u u u x u u x du u dx u x Let x x dx I Example + + - - - = + + - + - = + + - + - = + - + - = + - + = + = + = = = = = = = = + = 1 ln 6 6 3 2 1 ln 6 6 3 2 ] 1 ln 2 3 [ 6 ] 1 1 1 [ 6 1 ] 1 ) 1 [( 6 1 6 6 , ) ( , ) ( 6 ) 1 ( 6 6 3 2 3 2 3 2 3 3 2 3 5 6 2 6 3 3 6 5 6 3 3 1 2 1
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c x c u u du u u du u I du u dx u x u x u x Let x x dx I Example + - = + = + = + = = + = = - = - - + - = ) 1 arctan( 2 arctan 2 1 2 2 2 1 1 1 ) 1 ( 1 ) 2 ( 2 3 2 2 3
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Examples II Dealing with “rational functions of sinx and cosx”
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2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 2 2 2 2 2 2 2 1 1 1 1 1 sin cos cos 1 2 1 1 1 2 cos sin 2 sin 1 1 sin , 1 1 sec 1 cos , 1 2 arctan 2 arctan tan : 2 u u u u u x u u u u u x u u u u du dx u x u u on substituti the use We x x x x u x x x x x + - = + - + = - = + = + + = = + = - = + = = + = = = = +
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2 1 1 2 2 2 2 2 2 2 1 1 sin , 1 1 sec 1 cos , 1 2 arctan 2 arctan tan : 2 u u u u du dx u x u u on substituti the use We u x x x x x + = - = + = = + = = = = +
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2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 sin cos cos 1 2 1 1 1 2 cos sin 2 sin : u u u u u x u u u u u x on substituti the use We x x x x + - = + - + = - = + = + + = =
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+ - = + + - + = + + - = + - = - + - = + = + = = = = - = 3 10 3 2 1 3 10 3 1 2 1 3 10 3 1 2 5 3 sin 5 3 1 1 cos & 1 2 sin 1 2 arctan 2 arctan tan sin 5 3 ) 1 ( 2 2 2 2 2 2 2 2 2 2 2 2 2 u u du u u u u du I u u u u u x u u x u u x u du dx u x u u Let x dx I Example x x
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c c c u u du u u u u du I x x x x + - - = + - + - - = + - +
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