2. QUESTION ABOUT INTEGRALS OF THE FORM ( u2 + c )k du When using partial fractions to integrate a rational function, the one case not discussed in the class notes was how to integrate functions of the form Turk. In this question we will look at these types of integrals, and use them to integrate some rational functions. Suppose that k is a positive integer. (a) Assuming that k > 2, show that 2k - 3 da- 2k - 2 (u2+ C)k-17 2k - 2 J ( u2 + c )k - I du ) = - ( 2 2 + C )K (b) If c # 0 conclude that 1 2 k - 3 ( 2 2 + c ) k du = (2k -2)c (u2 + c)k-1 (2k - 2)c ) (u2 + c)k-I du The formula in (b) is an example of a reduction formula. It does not solve the an- tiderivative problem directly, but lets us reduce the problem of finding an antiderivative of Tuz tok to the problem of finding an antiderivative of (u2+c)k=1. By repeatedly applying the formula, we can reduce to the case k = 1, where we already know the antiderivative. (c) Find 1 (U2 + 9)2 (d) Find (u2+9)3" du. (e) Find 3x2 + 2x+ 9 - dx = 2x - 18 3 (202 + 9) 2 (12 +9)2 -dx. x2 + 9