362 Chapter 4 Definite Integrals 11. Write out all the integration formulas and rules that we 18. Consider the function know at this point. 12. Show that F(x) = sinx-x cosx+2 is an antiderivative of f(x)= =xsin x. ½) 13. Show that Fox) = x²-1) is an antiderivative of f(x)=x³ex²). 14. Verify that cotx dx = In(sin x) + C. (Do not try to solve the integral from scratch.) 15. Verify that Inx dx = x(lnx-1)+ C. (Do not try to solve the integral from scratch.) 16. Show by exhibiting a counterexample that, in general, dxf). In other words, find two functions f f(x) 8(x) 8(x)dx and g such that the integral of their quotient is not equal to the quotient of their integrals. 17. Show by exhibiting a counterexample that, in general, ff(x)g(x) dx (ff(x) dx)(g(x) dx). In other words, find two functions ƒ and g so that the integral of their product is not equal to the product of their integrals. In the definition of the definite integral we required integrands to be continuous. If an integrand fails to be continuous every- where, then a kind of branching can occur in its antideriva- tives. In Exercises 18-19 we investigate some discontinuous examples where we choose nonstandard antiderivatives. Outside of this set of examples we will restrict our attention to the standard antiderivatives in Theorems 4.16-4.19. oblies ris mor F(x)= In x, if x <0 In x+4, if x>0. Show that the derivative of this function is the function Compare the graphs of F(x) and In (x), and f(x) = discuss how this exercise relates to the second part of Theorem 4.16. 19. Consider the function f(x) F(x) -cotx, if x <0 -cotx+100, if x > 0. Show that the derivative of this function is the function = csc² x. Compare the graphs of F(x) and -cot and discuss how this exercise relates to the fourth part of Theorem 4.18. 20. Consider the function F(x) sec-1x,-л secx+x, ifx>1. if x<-1 Show that the derivative of this function is the function Compare the graphs of F(x) and sec f(x) = 1 and discuss how this exercise relates to the second part of Theorem 4.19. Skills Use integration formulas to solve each integral in Exer- cises 21-62. You may have to use algebra, educated guess- and-check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating. (Hint for Exercise 54: tanx= sinx COSX (Exercises 59-62 involve hyperbolic functions and their inverses; see Section 2.6.) 35. 3 x 3 tan² x dx 36. (sin x+ sin² x + cos²x) dr x-1 37. л dx 38. dx 1-x 7 dx 39. dx 40. √1-15 1+16x2 41. 1 dx dx 42. 13x19x2-1 4+x2 21. ((x²-3x³-7) dx 22. √(x³ + 4)² dx 2x 43. dx 44. 1+x2 12 sin 2 sin x cos x dx 23. f(x²-1)(3x+5) dx 24. (+1)山 dx 45. x²+2 2xe) dx 46. 25. (x² +23 +2²) dx 3 6x 26. dx 47. dx 48. x²+1 3x2+1 27. √x+1 dx 28. SG-4) e3x-2e4x dx 49. dx e2x 29. 0.203.437-50) de 30. fs 5 sin(2x+1) dx 51, x²(x³ + 1)³ dx 31. 4(ex-3y² dx 32. 42 4e2x-3 dx 53. 2x ln x-x 33. dx 3x2's 3x² sin(x3 + 1) dr x33x-2 50./sec² x + csc²x dx 52. 54. (tanx+xsec²x)dx Sta tan x dx sec(3x) tan(3x) dx (In x)2 34. 3 csc(xx) cot(xx) dx 55. (+税) dx 56. dx 2√√