121616949-math.180 - Start with the product rule d dx f x g...

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166 Chapter 8 Techniques of Integration Exercises Find the antiderivatives. 1. Z csc x dx 2. Z csc 3 x dx 3. Z p x 2 - 1 dx 4. Z p 9 + 4 x 2 dx 5. Z x p 1 - x 2 dx 6. Z x 2 p 1 - x 2 dx 7. Z 1 1 + x 2 dx 8. Z p x 2 + 2 x dx 9. Z 1 x 2 (1 + x 2 ) dx 10. Z x 2 4 - x 2 dx 11. Z x 1 - x dx 12. Z x 3 4 x 2 - 1 dx We have already seen that recognizing the product rule can be useful, when we noticed that Z sec 3 u + sec u tan 2 u du = sec u tan u. As with substitution, we do not have to rely on insight or cleverness to discover such antiderivatives; there is a technique that will often help to uncover the product rule.
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Unformatted text preview: Start with the product rule: d dx f ( x ) g ( x ) = f ± ( x ) g ( x ) + f ( x ) g ± ( x ) . We can rewrite this as f ( x ) g ( x ) = Z f ± ( x ) g ( x ) dx + Z f ( x ) g ± ( x ) dx, and then Z f ( x ) g ± ( x ) dx = f ( x ) g ( x )-Z f ± ( x ) g ( x ) dx. This may not seem particularly useful at first glance, but it turns out that in many cases we have an integral of the form Z f ( x ) g ± ( x ) dx...
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