121616949-math.184 - high degree polynomials in the...

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170 Chapter 8 Techniques of Integration Exercises Find the antiderivatives. 1. Z x cos x dx 2. Z x 2 cos x dx 3. Z xe x dx 4. Z xe x 2 dx 5. Z sin 2 x dx 6. Z ln x dx 7. Z x arctan x dx 8. Z x 2 sin x dx 9. Z x sin 2 x dx 10. Z x sin x cos x dx 11. Z arctan( x ) dx 12. Z sin( x ) dx 13. Z sec 2 x csc 2 x dx A rational function is a fraction with polynomials in the numerator and denominator. For example, x 3 x 2 + x - 6 , 1 ( x - 3) 2 , x 2 + 1 x 2 - 1 , are all rational functions of x . There is a general technique called “partial fractions” that, in principle, allows us to integrate any rational function. The algebraic steps in that technique are rather cumbersome if the polynomial in the denominator has degree more than 2, and the technique requires that we factor the denominator, something that is not always possible. However, in practice one does not often run across rational functions with
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Unformatted text preview: high degree polynomials in the denominator for which one has to find the antiderivative function. So we shall explain how to find the antiderivative of a rational function only when the denominator is a quadratic polynomial ax 2 + bx + c . We should mention a special type of rational function that we already know how to integrate: If the denominator has the form ( ax + b ) n , the substitution u = ax + b will always work. The denominator becomes u n , and each x in the numerator is replaced by ( u-b ) /a , and dx = du/a . While it may be tedious to complete the integration if the numerator has high degree, it is merely a matter of algebra....
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