76. Proof Prove each formula by mathematical induction. (You may need to review the method of proof by induction from a preealculus text.) (a) 221' = n(n +1) (1,) 213:].1202 +1): i=1 75. Finding Values Find the constants a and b that maximize the value of (1 - x2) dx. Explain your reasoning. 76. Step Function Evaluate, if possible, the integral [x ] dx. 77. Using a Riemann Sum Determine lim [12 + 22 + 32 + . . . + n]] 1-00 n by using an appropriate Riemann sum.113. Analyzing a Function Show that the function 1/x f (x) = dt + dt Jo 12 + 1 12 + 1 is constant for x > 0. 114. Finding a Function Find the function f(x) and all values of c such that f (t) dt = x2 + x - 2.97. Rewriting Integrals Assume that f is continuous everywhere and that c is a constant. Show that cb f(x) dx = c f(cx) dx. ca 98. Integration and Differentiation (a) Verify that sin u - u cosu + C = usin u du. TT 2 (b) Use part (a) to show that sin x dx = 217. o 99. Proof Complete the proof of Theorem 4.16. 100. Rewriting Integrals Show that if f is continuous on the entire real number line, then Pb+ h f(x + h ) dx = f(x) dx. a Jath10 10 8 8 -.i.-. 6 .... 4 2 2 X X 4 5 6 8 10 Figure for 35 Figure for 36 36. Finding the Area of a Region Approximate the area of the shaded region using (a) the Trapezoidal Rule with n = 8. (b) Simpson's Rule with n = 8