M23 Techniques of Integration

# M23 Techniques of Integration - TECHNIQUES OF INTEGRATION...

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TECHNIQUES OF INTEGRATION Integration by Parts udv uv vdu Note: u and v must be chosen from the integrand in such a way that the resulting integral vdu is easier to evaluate than udv . Powers of Trigonometric Functions Type I: du u u n m cos sin Case 1 . Either m or n is a positive odd integer. If m is odd, factor out sin udu and express the remaining factor of sine to powers of cosine by using sin cos 2 2 1 u u . If n is odd, factor out cos udu and express the remaining factor of cosine to powers of sine by using cos sin 2 2 1 u u . Case 2 . Both m and n are positive even integer or one is positive even integer while the other is 0. Reduce the power by using the identities cos cos 2 1 2 2 u u 2 1 cos2 sin 2 u u Type II: tan sec m n u udu Case 1 . n is an even positive integer. Factor out 2 sec udu and express the remaining factors of secant in terms of tangent by using 2 2 sec 1 tan u u . Case 2 . m is an odd positive integer.

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