Review of Integration Techniques

Review of Integration Techniques - 1 Review of Integration...

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1 Review of Integration Techniques Math 192 - Spring 1997 Written by Don Allers; Revised by Sean Carver 1 Methods of Integration Substitution Idea: Rename an ugly piece of the integrand and see if it reduces to a simpler form that you can integrate. When? Try this method if your integral has ugly pieces like e tan v or cos(ln x ), or if it looks like substitution will turn the integral into an easy one, such as R x n dx , R e x dx , or any of the other known integrals in Table 7.1 on page 556. Eg: (a) Z e tan v sec 2 v dv, (b) Z dx x cos(ln x ) , (c) Z 2 w dw 2 w , (d) Z dx x ( x + 1) . Integration By Parts When? R f ( x ) g ( x ) dx where f ( x ) can be easily differentiated, and g ( x ) easily integrated. Good choices for f ( x ) are f ( x ) = a 0 + a 1 x + · · · a n x n (any polynomial), f ( x ) = ln( x ), or f ( x ) = tan - 1 ( x ) (or any other inverse trig function). Often g ( x ) is either sin( x ), cos( x ), or e x , but g ( x ) can be anything if f ( x ) is a log function or an inverse trig function.
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