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Unformatted text preview: CALCULUS II Spring 2009 LECTURE 1 This lecture provides examples of trigonometric, logarithmic and exponential integrals. It introduces • R cos 2 ( x ) dx , R cos 3 ( x ) dx , etc. • R tan 2 ( x ) dx , R tan 3 ( x ) dx , etc. • R sin 2 ( x )cos 2 ( x ) dx , R sin 3 ( x )cos 3 ( x ) dx , etc. • R sin 2 ( x )cos 3 ( x ) dx , etc. • the notion of integration by parts • R ln( x ) dx • R x cos( x ) dx • R xe x dx . 1 Recall We recall from Calculus I • the integration technique called usubstitution Z u ( d ) u ( c ) f ( u ) du = Z d c f ( u ( t )) u ( t ) dt • the integral formulas R sin( x ) dx = cos( x ) + C R cos( x ) dx = sin( x ) + C R tan( x ) dx = ln(  cos( x )  ) + C End of Recall 2 To date we have investigated integrals of the six main trigonometric func tions, sin( x ), cos( x ), etc. and the exponential function e x , etc. Of interest, now, are integrals of functions of the form, e.g., sin 3 ( x ) and sin 3 ( x )cos 2 ( x ). Further, the integral of the natural logarithmic function, ln( x ), remains to be investigated. Functions of the form x sin( x ) and xe x can be integrated using the method of integration by parts . We attend to these matters in this lectures. Lectures 2 and 3 deal with additional func tions not mentioned here. One might think that there are an infinite number of function types whose integrals are interesting. So why are these notes focused on a select few? One answer is that, roughly speaking, the integrations selected for these notes comprise most of what we know how to do with pencilandpaper. It is true that in practice most occurring integrals are sufficiently involved that there is no hope for a pencilandpaper computation. Instead one rou tinely uses numerical schemes. However, pencilandpaper integrations are possible frequently enough that they are worth studying (independently of numerical schemes). To help anchor all of this one notes that there is no known closed form ex pression for the indefinite integral R sin( x 2 ) dx . Mathematics Interlude I Integration of Trigonometric Functions Rather than trying to identify a “complicated” general approach to inte grating trigonometric functions we illustrate key ideas through examples. EXAMPLE. Evaluate the integral Z π 2 sin( x ) dx. 3 Solution. One writes Z π 2 sin( x ) dx = cos( x )  π 2 = [0 1] = 1 . Thus R π 2 sin( x ) dx = 1. EXAMPLE. Evaluate the integral Z π 2 cos 2 ( x ) dx. Solution. One writes Z π 2 cos 2 ( x ) dx = Z π 2 1 2 + cos(2 x ) 2 dx = x 2 π 2 + Z π 2 cos(2 x ) 2 dx = π 4 + sin(2 x ) 4 π 2 = π 4 + sin( π ) 4 sin(0) 4 = π 4 . cos ( x + y ) = cos ( x ) cos ( y ) sin ( x ) sin ( y ) cos (2 x ) = cos 2 ( x ) sin 2 ( x ) = 2 cos 2 ( x ) 1 cos 2 ( x ) = 1 2 + cos (2 x ) 2 Thus R π 2 cos 2 ( x ) dx = π 4 ....
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 Spring '08
 ECKER
 Integrals, Formula One, π π

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