Show that (1). 1 x 3 x 5 x .. where k is a positive integer. 2* X (2k - 1) 2k + 2 1x3x5xx( where k is a positive integer. 30. Show that x"dx 1-x 2 (2k-1) 2 1x3x5xx (n-1) 2x4x6x xn if n is an even positive integer. 2 31. show that x"dx 2x4x6xx (n-1) 1 x 3 x 5x xn if n is an odd positive integer. 32. Show that S dx (1/n) n F(1/n+) 33. Evaluate the following definite integrals: (a) -x4 dx. (b) S4x*e** dx. 0 sin x dx. cos x 34. Show that r() r() Vtan 0 de- 2 35. Show that d" F(y) dy" S-e-* (logy)" dx 36. Show that by a suitable change in variable we have (n) a-l (m) - S. (log: 1) dy 37. Evaluate the integral by expanding the integral in series, and show that x' + x -R Sedn-x-3x 15 x 2! 7x 3! 9x4! where R 38. Show by integrating by parts that of x. x 3 1 x 3 x 5 dn + + 2 2x 2x2 22x4 23 (-1)***** Show how this expression may be used to compute the value of erf(x) for large values