x! m in VS .NET

Maker QR Code JIS X 0510 in VS .NET x! m

x! m
QR Code 2d Barcode Reader In VS .NET
Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in .NET framework applications.
Quick Response Code Generation In .NET Framework
Using Barcode maker for .NET Control to generate, create QR Code image in .NET applications.
15.36. Prove that if m is a positive interger, m 1 2 1
Decoding Denso QR Bar Code In Visual Studio .NET
Using Barcode decoder for .NET Control to read, scan read, scan image in .NET applications.
Barcode Creator In Visual Studio .NET
Using Barcode printer for Visual Studio .NET Control to generate, create bar code image in VS .NET applications.
p 1 m 2m  1 3 5 2m 1
Barcode Scanner In .NET
Using Barcode decoder for Visual Studio .NET Control to read, scan read, scan image in .NET framework applications.
QR Code ISO/IEC18004 Printer In Visual C#.NET
Using Barcode maker for .NET Control to generate, create QR Code image in VS .NET applications.
15.37. Prove that 0 1
QR Code Creator In Visual Studio .NET
Using Barcode generation for ASP.NET Control to generate, create QR Code 2d barcode image in ASP.NET applications.
QR Code ISO/IEC18004 Creator In Visual Basic .NET
Using Barcode creation for Visual Studio .NET Control to generate, create QR Code ISO/IEC18004 image in .NET framework applications.
e x ln x dx is a negative number (it is equal to
Barcode Creator In .NET
Using Barcode generator for .NET framework Control to generate, create barcode image in .NET framework applications.
UCC - 12 Printer In .NET Framework
Using Barcode generation for VS .NET Control to generate, create UCC - 12 image in .NET framework applications.
, where
GS1 - 13 Generator In .NET Framework
Using Barcode drawer for VS .NET Control to generate, create EAN13 image in .NET applications.
Encode Leitcode In VS .NET
Using Barcode creator for .NET framework Control to generate, create Leitcode image in .NET applications.
0:577215 . . . is called
Make Bar Code In Visual Basic .NET
Using Barcode generation for .NET Control to generate, create bar code image in VS .NET applications.
EAN-13 Decoder In None
Using Barcode reader for Software Control to read, scan read, scan image in Software applications.
Euler s constant as in Problem 11.49, Page 296). THE BETA FUNCTION 15.38. Evaluate (a) B 3; 5 ; 1 15.39. Find Ans: (a)
Code 128C Printer In None
Using Barcode generation for Font Control to generate, create Code-128 image in Font applications.
Code 39 Decoder In Java
Using Barcode recognizer for Java Control to read, scan read, scan image in Java applications.
b B 3=2; 2 ; b
Read ECC200 In Visual Basic .NET
Using Barcode reader for Visual Studio .NET Control to read, scan read, scan image in Visual Studio .NET applications.
Matrix 2D Barcode Encoder In Java
Using Barcode creation for Java Control to generate, create 2D Barcode image in Java applications.
c B 1=3; 2=3 : c 2
Generating Barcode In None
Using Barcode generation for Microsoft Word Control to generate, create bar code image in Microsoft Word applications.
GTIN - 12 Generation In .NET
Using Barcode creation for Reporting Service Control to generate, create UPC-A Supplement 2 image in Reporting Service applications.
Ans:
a 1=105;
b 4=15;
p c 2= 3
x2 1 x 3 dx;
1 p 1 x =x dx;
4 x2 3=2 dx.
a 1=60; b =2; c 3 3 4 dx p : u3=2 4 u 5=2 du; b 15.40. Evaluate (a) 3x x2 0 0 a 15.41. Prove that dy f 1=4 g2 p : p 4 y4 4a 2 0 a =2 15.42. Evaluate (a)
Ans:
a 12;
b 
sin4  cos4  d;
2
cos6  d:
Ans:
a 3=256;
b 5=8
 15.43. Evaluate (a)
sin5  d;
=2
cos5  sin2  d:
Ans:
a 16=15;
b 8=105
15.44. Prove that
=2 p p tan  d = 2.
CHAP. 15] 1 15.45. Prove that (a)
GAMMA AND BETA FUNCTIONS x dx  p ; 1 x6 3 3 1 b
y2 dy  p . 1 y4 2 2
1 15.46. Prove that
e2x 2 dx p 2=3 1=3 where a; b > 0. ae3x b 3 3a b 1 e2x 2 dx p 1 9 3
1 15.47. Prove that
e3x
[Hint: Di erentiate with respect to b in Problem 15.46.] 15.48. Use the method of Problem 12.31, 12, to justify the procedure used in Problem 15.11. DIRICHLET INTEGRALS 15.49. Find the mass of the region in the xy plane bounded by x y 1; x 0; y 0 if the density is  Ans: =24 15.50. Find the mass of the region bounded by the ellipsoid p xy.
x2 y2 z2 1 if the density varies as the square of a2 b2 c2
the distance from its center. abck 2 Ans: a b2 c2 ; k constant of proportionality 30 15.51. Find the volume of the region bounded by x2=3 y2=3 z2=3 1. Ans: 4=35 15.52. Find the centroid of the region in the rst octant bounded by x2=3 y2=3 z2=3 1. " " " Ans: x y z 21=128 15.53. Show that the volume of the region bounded by xm ym zm am , where m > 0, is given by 8f 1=m g3 3 a . 3m2 3=m
15.54. Show that the centroid of the region in the rst octant bounded by xm ym zm am , where m > 0, is given by " " " x y z 3 2=m 3=m a 4 1=m 4=m
MISCELLANEOUS PROBLEMS b 15.55. Prove that x a p b x q dx b a p q 1 B p 1; q 1 where p > 1; q > 1 and b > a. [Hint: Let x a b a y:] 3 15.56. Evaluate Ans: (a) dx p ; x 1 3 x 1 b 7 p 4 7 x x 3 dx.
a ;
2 f 1=4 g2 p b 3 
15.57. Show that
p p 32 f 1=3 g2 p . 1=6 3 1 2 1 xu 1 xv 1 u v dx where u; v > 0. 0 1 x
15.58. Prove that B u; v
[Hint: Let y x= 1 x :
15.59. If 0 < p < 1 prove that 1 15.60. Prove that =2
GAMMA AND BETA FUNCTIONS  p sec . 2 2
[CHAP. 15
tan p  d
xu 1 1 x v 1 B u; v u where u; v, and r are positive constants. r 1 r u v x r u v 0
[Hint: Let x r 1 y= r y .] =2 15.61. Prove that
sin2u 1  cos2v 1  d B u; v where u; v > 0. 2av bu a sin2  b cos2  u v
[Hint: Let x sin2  in Problem 15.60 and choose r appropriately.] 1 15.62. Prove that dx 1 1 1 xx 11 22 33 0
15.63. Prove that for m 2; 3; 4; . . . sin  2 3 m 1  m sin sin sin m 1 m m m m 2
[Hint: Use the factored form xm 1 x 1 x 1 x 2 x n 1 , divide both sides by x 1, and consider the limit as x ! 1.] =2 15.64. Prove that
ln sin x dx =2 ln 2 using Problem 15.63.
[Hint: Take logarithms of the result in Problem 15.63 and write the limit as m ! 1 as a de nite integral.] 15.65. Prove that         1 2 3 m 1 2 m 1 =2 p : m m m m m
[Hint: Square the left hand side and use Problem 15.63 and equation (11a), Page 378.] 1 15.66. Prove that
ln x dx 1 ln 2 . 2
[Hint: Take logarithms of the result in Problem 15.65 and let m ! 1.] 1 15.67. (a) Prove that
sin x  ; dx xp 2 p sin p=2
0 < p < 1.
(b) Discuss the cases p 0 and p 1. 1 15.68. Evaluate Ans: a (a)
0 1 2
1 sin x2 dx; b b
x cos x3 dx.
p =2;
 p 3 3 1=3 0 < p < 1.
1 15.69. Prove that
x p 1 ln x dx 2 csc p cot p; 1 x p ln x 2 2 . dx 16 x4 1
1 15.70. Show that
15.71. If a > 0; b > 0, and 4ac > b2 , prove that 1 1
1 1
e ax
bxy cy2
2 dx dy p 4ac b2
CHAP. 15]
GAMMA AND BETA FUNCTIONS
15.72. Obtain (12) on Page 378 from the result (4) of Problem 15.20. p 3 [Hint: Expand ev = 3 n in a power series and replace the lower limit of the integral by 1.] 15.73. Obtain the result (15) on Page 378. 1 [Hint: Observe that x x ! , thus ln x ln x 1 ln x, and x 0 x 0 x 1 1 x x 1 x Furthermore, according to (6) page 377. x ! lim k! kx k!1 x 1 x k
Now take the logarithm of this expression and then di erentiate. Also recall the de nition of the Euler constant,
. 15.74. The duplication formula (13a) Page 378 is proved in Problem 15.24. positive integers, i.e., show that p 22n 1 n 1 n 2n  2 For further insight, develop it for
Hint: Recall that 1 , then show that 2   2n 1 2n 1 5 3 1 p n 1 : 2 2 2n Observe that 2n 1 2n ! 2n 1 5 3 1 2n n 1 2n n! Now substitute and re ne.
Copyright © OnBarcode.com . All rights reserved.