Lecture 8.pdf - LECTURE 8 INTEGRATION OF TRIGONOMETRIC...

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LECTURE 8, INTEGRATION OF TRIGONOMETRIC FUNCTIONSPO-NING CHENIn this lecture, we will learn how to integrate powers of trigonometric functions such asZsin4xcos5xdxor in general, integrals of the formZsinmxcosnxdxwheremandnare positive whole numbers. Ifmornis odd, this can be computed usingsubstitution. Otherwise it is more complicated.Example 1ComputeZcos4xsinxdx.Answer to Example 1We use the substitutionu= cosxWe have thendu=-sinxdxandZcos4xsinxdx=-Zu4du=-u55+c=-cosx55+CIn fact, we can compute the integral in very similar way as long as one of the wholenumber is odd since we can usecos2x+ sin2x= 1Example 2ComputeZcos2xsin3xdx.Answer to Example 2We use the substitutionu= cosxWe have thendu=-sinxdx1
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