Bài tập tích phân và ứng dụng

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Tài liệu tham khảo - Hệ thống bài tập tích phân - ứng dụng của tích phân

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Nội dung Text: Bài tập tích phân và ứng dụng

x2 x 1 x4 .dx ; .dx 2 x 2 2x 1 x 2x 1 x2 1 dx .dx ; .dx 3x 2 4 x4 1 dx dx .dx ; .dx x 1 sin x sin x g ( x) sin x.dx dx 2 x 1 .dx ; .dx x. ln x. ln(ln x) cos 2 x 1 f ( x) ex x 2x 3 x ..dx e 2 ..dx ; 2 1) 3 (x ex dx ex. 2 ..dx ; cos 2 x x. ln x x x2 g ( x) a , a #0 2 x .3 x (e x 1) 3 .dx ; dx 9x 4x x2 f ( x) a , a #0 sin 2 x. cos x..dx ; cot gx.dx 2 h( x ) ( x 2) x a , a #0 dx dx (sinx cosx).dx ; ; F ( x) x ln(1 x) 5 1 cos x cos x sinx - cosx x f ( x) 1 x x2 a 4x 2 6x 1 x2 F ( x) x a ln x a , a #0 2 3x 3 f ( x) 2; f(x) 2 2 2x 1 x2 f ( x) a 2 x 4 3x 2 1 2 f(x) ; f ( x) 3 2 x x x6 x 2 ( x ln x 1) 3 1 4x 9x 1 khi x 0 f(x) ; f ( x) F ( x) 4 2 4x 2 9 x x2 0 khi x 0 x.lnx khi x 0 3 x4 4 x x4 x ; f ( x) f(x) x 2 f ( x) 0 khi x 0 1 1 f ( x) ; f ( x) 2x 2x 1 4x x3 3 2 F ( x) (ax bx c) 2 x 3 voi x 2 2 32 x 2x ; 2 2 x .33 x .4 4 x f ( x) f(x) 2 20 x 30 x 7 f ( x) 2x 1 5x 1 e3x 2 ; 2x 3 f ( x) f(x) 10 x x2 x.(1 x)10 .dx ; dx (1 x)100 3 1 1 1 x dx dx x.dx x. 2 5 x .dx ; dx 3 x x x 3 1 3x 24 x )( x 4 (x x x ).dx
dx dx 3x 2 3x 3 A ;B y x 3 3x 2 x2 2x. 2x 1 5x 6 x 3 dx dx A ;B a b c 3 1 x2 y ( x 1).(2 x) ( x 1) 2 ( x 1) ( x 2) dx dx A ;B 2 1 x x1 ( x 1) x 2x 2 4 4 4 f ( x) cos x ; f(x) sin x cos x (6x 3 x 2 2dx 8x 1)dx A ;B 6 6 2 f ( x) cos x sin x ; f(x) cot g x x2 1 (3x 2 4). x 2 1 1 dx 8 cos 2 x. sin 3 x ; f ( x) f(x) x 3 .3 x 2 1.dx ; B A sin 4 x 2 x x1 1 cos 2 x f ( x) ; f(x) 3 cos 2 x. sin 2 x cos x. sin x 2dx cos x sin x. cos x A ;B dx sin x cos x x f ( x) ; f(x) 2 sin x cos x 1 2 sin x x 3x 2 2 4 3 sin 2 x dx 1 A ; B dx 1 1 sin x. cos3 x f ( x) ; f(x) sin 2x 2 sin x x x3 x 2 1) 2 (x dx sin x A ;B dx 1 x1 3 5 cos x sin 2 x 1 4 sin x. cos x f ( x) ; f(x) 1 ex x.(1 x.e x ) x2 x 3 (1 5x 2 )10 dx; B A dx 2x 2 2 2 x 2 x 13 x xe dx dx 3x 2 f ( x) ; f(x) A dx; B dx x3 x2 (4 x 2 ) 3 (4 x 2 ) 3 x2 x1 1x f ( x) ; f(x) 1 x 6 .dx x 5 dx x3 A ;B ; 2 1- x x 1 x2 1 2x f ( x) ; f ( x) ; x 2 dx x2 1 x 1x A ; x x2 2 x1 x 2 . a x .dx B .dx A x1 x 3 dx sin x. cos3 x.dx sin 2 x (2x 1).dx A ;B A ;B dx ( x 8 4) 2 4 2x 3 3x 2 2x 3 1 cos2 x cos6 x x 1 x2 1 x2 3 cos5 x. sin x.dx; B A dx A dx ; B .dx e ex/ 2 x x4 1 4 3x 2 2) x( x 1 x x (1 ln x).dx; B A dx 1 x4 1 e 4e x x A dx ; B .dx 6 2 x( x 4 1) x( x 1) xdx 1 x 2 .dx A ;B 2 1 x2 . 1 1 x2 ln x x 2 sin 2 x f ( x) ln x ; f(x) ; f(x) dx dx x A ;B .dx 2x x 1. ( x 1) 2 1 ( x 1) 2 .cos 2 x ; f(x) x 2 1 e 2x 1 ; 3 3 1e f ( x)
e 2 x .sinx ; f(x) e -2x . cos 3 x; 1 1 f ( x) f ( x) ; f ( x) 2 2 3x 2 x 1 x 2x 2 (cot g 2 x cot gx 1)e x ; f ( x) 1 1 f ( x) ; f ( x) 2 2 ( x 2 x 2) 3 2 (3 x 2 x 1) ax A x. cos x .dx; B e . sin(bx).dx 7 x 13 7 x 13 f ( x) ; f ( x) e 2 x . cos2 x.dx; B x n . ln x.dx A 2 ( x 4 x 5) 3 2 ( x 4 x 5) x 2 .e 3x .dx; B x 2 . sin(3x).dx A x2 2x 3 x1 f ( x) : f(x) 2 x3 1 2 x x2 x .e dx x 2 . cos(2x).dx A ;B 3 ( x 2) 2 x 1 f ( x) ; f(x) 2 x(x 1) 2 x 2x 1 (1 sin x)e x .dx ln(sin x) A .dx; B sin 2 x 1 cos x x.dx x A ;B .dx eax.sin(bx).dx A x.cos x .dx; B x 2x 2 1 4 3 x 3x 2 x.5 dx x5 (x3 4x 2 2x 7).e 2 x .dx; A A ;B .dx x6 x3 2 x8 1 (1 x 7 ).dx x4 A ;B .dx dx x A ; B .dx x( x 7 1) ( x10 10) 2 sin3 x cos2 x cos2 x 1 x A x. ln .dx; B .dx ( x 3 1).dx x3 sin3 x 1 x A ;B .dx x 3 5x 2 6 x ( x 1)100 x.dx x 2 1).dx A ; B ln(x ( x 2 1).dx x 2 4x sin 2 x A ;B .dx x4 x3 x 2 x 1 x3 4 x 2 5x 2 x4 x 2 1 sin 2 f ( x) a) f ( x) b) f ( x) x3 3 2 x x x 5 f ( x) cot g 6 x; f ( x) tg x; f ( x) cos3 x.sin8x; f ( x) cos3 x.sin2 x; 1 f ( x) cos x. cos2x. sin 4x; f ( x) x( x 1) 2 f ( x) cos x. cos2x. cos3x 3x 2 3x 3 (1 sin x)dx cos x. sin x.dx y A ;B x 3 3x 2 sin x(1 cos x) sin x cos x dx cos x.dx A ;B a b c sin x cos x 1 13 10sin x cos 2x y ( x 1) 2 ( x 2) ( x 1) dx A ; sin x sin 2x cos2 x 2 dx B 3 sin x 8 sin x. cos x 5 cos2 x 2 2001 x sin 2x.dx cos 2x.dx f ( x) A ;B 2 1)1002 (x 2 sin 4 x cos4 x sin x 1 dx dx A ;B 2 4 sin x. cos5 x 3 sin x. cos x
(3 2 x 2 x ) 2 ; F(x) 2 2x .33 x .4 x (sin x cos x)dx dx F ( x) A ;B cos3 x sin x 2 cos x ex e 3 x 2 : F(x) F ( x) cos4 x.dx (sin x sin3 x).dx ex e x A ;B sin3 x 2 cos2 x 1 2 5x 2x 1 5x 1 e 1 F ( x) : F(x) (cos x sin x).dx dx ex 10 x A ;B 1 sin 2x sin 2x 1 (x2 x 1).e x (x - 1).e x F ( x) : F(x) x2 x2 1 e ax . sin(bx).dx; B e 2 x . sin 2 x.dx A x 3 .dx x n . ln x.dx; B x 2 .e 3x dx A x3 x 4 .dx; B A x4 2x 2 1 x 2 . ln(2x 1).dx A sin(ln x).dx; B 2 dx (x x x 1)dx (2 x 3 5x 2 2x 4).e 2 x .dx; A A ;B 2 2 x x x1 x x x11 2.e x .dx ln(sin x)dx (4x 5).dx dx A ;B A ;B sin 2 x 1 ex x 2 6x 1 (1 x 2 ) 3 (1 sin x).e x dx ln(cosx).dx A ;B cos2 x 1 cos x dx dx 1 1x A ;B A . ln .dx; 2 ( x 1). 3 2x x 2 ( x 1) 1 x 2 1x 1x dx A ; x 2 1)dx 2x 1 2x 3 dx x. ln(x A ;B dx 1 ex . x2 1 B (2x 1) 2 3 2x 1 ln x.dx ex x A ;B e 2.dx x. ln x 1 dx x2 ln( x 3) C 2 x 3 x2 F ( x) 3.dx x F ( x) 3 2 x.dx 10 x1 ( x 3 1).dx; B A 2 -1 x 2 1 e2 1 5 7x 2 x 5 dx F ( x) tgx A .dx; B 2x 1 2x 1 x x2 x2 1 2 2 dx cos 3 x.dx 2 ( x 1).dx I ;B ; A x 2 x ln x x2 3 x1 sin x 1 6 1 ex x 4 tgx .dx e A ;B dx; cos 2 x x x 0e e 2 x 0 F ( x) (x 3 x 2).e 1 2 e x .dx dx A ;B ; x F ( x) 2 . cos( x )e ex x 4x 2 4 e 8x 0 1
ln 3 2 .dx dx 1 A ;B ; (1 3 x)(1 2 x 3 x 2 )10 .dx; I x x 1 sin x ee 0 0 0 1 x5 1 2 dx dx I .dx; A ;B ; x2 1 sin 4 x 2 xx 1 0 1 a 2 4 dx I ; 2 x2 )2 0 (a x 2 6x dx 3 dx 3 A ;B ; t 1 2 2 x 4x 2 0 sin x 3cos x 19 x 5 (1 x 3 ) 6 .dx; I 0 1 x.dx 2 2 I 2 4 x2 1 A cos 5 x. sin 3x.dx; B sin x. cos ( x )dx x 0 4 0 4 1 1 x2 x A .dx; B .dx; 3 4 4 x2 x1 2 0 0 A x 2 .dx; B x 3 x 2 .dx 1 0 1 x2 dx 3 -1 A .dx; B x2 2 x x1 1 2 F ( x) A. sin( x) B 2 1 1 F ( x).dx 4 A x. 1 x .dx; (DHTM - 1995) 0 0 1 F ( x) a. sin 2 x b. cos 2 x 1 x 2 .dx; (DHYHN 1998) A 2b F, 1 2 va a.dx 1 2 2 1 a (1 x 2 ) 3 .dx; (DHY HP 2000) A 0 4 4 x2 3 x 10 log 2 ( dx) dx 3 dx x5 A .dx; (HVQY 1998) 0 0 x. x 2 1 2 3 a b x 5 1 x 2 .dx; F ( x) 2 A x2 x 0 1 , F ( x) 4 va F(x).dx 2 - 3.ln2 2 3 tg 4 x.dx 1 A sin x .dx; B 2 cos 2 x 0 0 F ( x) a. sin 2 x b 2 3 2 dx tgx.dx F, 0 4 va F ( x).dx 3 A ;B 2 sin x cos x 1 cos x sin x. cos x 0 0 6 2 sin 2 x.dx I 4 0 1 sin x 1 x(1 x)19 .dx; A 0 2 cos x.dx I cos 2 x 0 11 7 sin x
4 2 sin x cos x 3 A dx ; B sin x sin x dx 3 2 sin x sin x . sin x cos x 0 0 I . cot gx.dx sin 3 x 2 3 4 4 sin x sin x 3 A dx ; B dx 6 3 cos x cos x x. sin x.dx 0 I 3 4 cos 2 x 09 tg 2 x 3 6 3 1 cos x A dx ; B dx 3 2 tg 2 x 4 sin x. cos x.dx 1 sin x I 0 1 cos 2 x 6 0 4 2 sin x cos x sin 2 x A dx ; B dx 6 cos x.dx 1 cos 2 x 2 sin 2 x I 0 0 5 sin x sin 2 x 06 e e 1 ln x dx x. sin x. cos 2 x.dx I A dx ; B x 2 x1 ln x 0 1 1 e4 e ln 2 x dx 3 dx (ln x ) 1 A ;B e 1 2 ln x .dx 1 2x x cos 2 (1 ln x ) x A ;B . ln .dx 1 1 e 2 2x 2x 4x 1 2 ln 2 dx dx 1 0 A ;B ln 2 dx x ex 1 e 1 0 ln 2 I ln 3 1 x x e 1 dx e dx 0 A ;B x x 1 e e x x dx e e 0 0 I e2x ex 0 e ln 13 e x dx ( x 1 ) dx 1 ln 2 e 2x 3e x A ;B x x (1 xe x ) A e .dx; B .dx ex) ex (3 1 e 2x 3e x 3 1 ln 5 0 0 1 2 1 1 x A ln dx ; 2 1 x 1 x 0 3 1 x 3 A .dx; B 1 x .dx; 3 4 dx dx x1 A dx ; B 0 0 sin x . cos 2 x cos x 4 cos 2 x sin 2 x 1 1 0 x 6 3 A x 1 x dx; B dx; x2 x1 0 1 2 1 x2 6 2x 4 x 2 dx; B A dx; x6 1 1 0 4 4 x dx e A ;B dx; 3 2 x2 x x 2 . cos x.dx x A x. cos x.dx; B 1 1 0 0 3 2 4 2 x.dx sin x e x . cos 3 x.dx A cot gx . dx ; B . dx A ; B sin 2 x 1 3 cos x 0 0 6 4 e 6 2 e 2 x sin 2 x.dx; B A cos(ln x).dx cos x A 1 4 sin x cos .dx; B e . cos x dx 2 0 0 0 0
1 ln 2 e x 3 sin 3 x 2 2 A x.e .dx; B ln x.dx 1x x 2 . ln A .dx; B .dx 0 1 1x 1 cos x 1 e 1 2 2 x. ln 2 x.dx; B x. ln( x 2 1).dx A 0 0 e 2 ln x cos 2004 x 2 2 sin 2 x 2 A (1 ln x) .dx; B .dx A .dx; B .dx 2 1x 4 2004 x sin 2004 x 0 1 sin x 0 cos 1 e2 1 1 x. sin x x. sin x A .dx; A .dx; B .dx ln 2 x ln x cos 2 x 2 03 0 1 cos x e 4 e sin 2 x.dx x ln x ) 2 dx A e dx ; B (1 A ; 3x 1 1 1 4 e 2 3 (x2 A x 1 ) ln x .dx ; B x . sin x . cos xdx A sin x. sin 2 x. sin 3 x. cos 5 x.dx; 1 0 0 2 3 2 x 2 ) dx ; B 2 A ln( x 1 cos ( x ) dx 3 A x. sin x.dx; B sin(sin x nx).dx 2 0 0 0 4 2 1 (x7 x5 x 3 x 1)dx 2 4 4 2 x sin x x 2 . sin 9 x.dx; B A sin x dx ; B dx A cos 4 x 1 cos x 0 1 3 2 4 e2 2 e ln(ln x ) ln x A dx ; B dx 1 x x 1 x2 e 1 I .dx 1 2x 1 2 2 x cos x x. cos 2 x.dx I I .dx 4 sin 2 x 0 1 2 e x sin 2 ( x).dx I x. sin 3 x.dx I 0 e 0 I (2 x 2). ln x.dx sin 2 x I .dx 1 3x 1 4 1 x4 5e x . sin 2 x.dx I I .dx 2x 11 0 2 e x . cos 2 x.dx I f (tgx) neu 0 x 0 2 g ( x) 2 I x . sin x.dx f (0) neu x 2 0 0; 2 1 2 x 5 cos 2 x.dx; B x 3 e x .dx A 1
4 2 3x 2 3x 3 A B C g ( x).dx g ( x).dx x 3 3x 2 ( x 1) 2 x1 x2 0 4 2 3x 3x 3 I .dx x3 3x 2 1 x 5 .dx 3 0 x 2 .dx dx I A ; B ; 2 0x 1 x) 9 2 2 (1 x 3x 2 1 2 4 (x2 x 3 dx 2 x 2.dx A ; B ; 2 (1 x 2 ).dx 1 x3 1 10 2 ( x 1) I HD : t x 1 x4 1 x 1 3 2 1 (2 x 10 x 16 x 1).dx A ; x 2 5x 6 3 1 x2 A B ( x 2) I .dx 1 dx ( x 1) 2 ( x 1) 2 2 x1 2 ( x 1) B ; 3) 2 ( x 1) 2 0 (x x f ( x) 1 0 3 2 ( x 1) ( x 1) 3 2 (x 3x x 6).dx (7 x 4)dx A ;B ; 3 2 3 x 5x 6 x 1x 3x 2 1 2 2 dx dx A ;B ; Ax 2 Bx C dx dx 3 2x 2 4 4x 2 x x x 3 f ( x)dx D E 1 1 ( x 1)( x 2) 2 x1 x1 2 1 (x3 x 2 4 x 1).dx x 3 .dx A ;B ; 3 x4 x3 8 4) 2 0 (x f ( x)dx 1 2 3 4 2 dx (1 x ).dx A ;B ; 6 2 4 x( x 1) 1 x.( x 1) 1 3 4 1 5 2x2 2x x dx 13 A ;B dx ; 3 2 dx tgx.dx x6 x3 ( x 2 )( x 2 1) 2 2 A ;B 3 0 3 2 1 sin x cos x cos x sin x. cos x 0 6 3 3x 2 2 I .dx 3 x2 tg 4 x.dx 3 01 A ;B ( cos x sin x ).dx cos 2 x 1 dx 0 I 6 x2 5x 6 0 4 2 1 ( x sin x)dx x sin 2 x. cos 2 2 x.dx A ;B I .dx 1 cos x 3 0 (1 2 x ) 0 0 1 3 2 (x 2 x 10 x 1).dx 2 x. cos x.dx I A ; x 2 2x 9 1 sin 2 x 0 0 1 (4 x 11).dx I 2 0x 5x 6 1 3.dx 2 2 sin 2 x.dx sin 2 x.dx I I ; va J x3 1 4 4 0 1 sin x 0 cos x 1 0 1 dx I 4 4x 2 x 3 sin x 0 f ( x) sin x cos x
sin 2 x.dx 3 cos x sin x I f ( x) AB cos 6 x cos x sin x 6 3 I f ( x).dx 2 1 sin 2 x cos 2 x. 0 I .dx sin x cos x 6 cos 4 x.dx sin 4 x.dx 2 2 4 3 4 sin 4 x 4 sin 4 x dx 0 cos x 0 cos x I x sin cos 4 x.dx 2 2 I 4 sin 4 x 2 0 cos x I 1 sin x .dx 0 2 dx I 2 1 sin 2 x 0 cos 3 x. sin 2 x.dx I 0 sin 3 x sin x 2 4 sin 4 x.dx I . cot gx.dx I sin 3 x 2 0 1 cos x 3 2 (3 sin x 4 cos x)dx I 2 4 cos 2 x 0 3 sin x 2 sin x. cos x.(1 cos x) 2 .dx I 0 3 dx I 4 dx sin x. sin x I 6 6 4 0 cos x 4. sin 3 x.dx 4 sin 2 x h( x ) I (2 sin x) 2 1 cos 4 x 0 A. cos x B. cos x h( x ) 2 cos 2 x.dx (2 sin x) 2 2 sin x I 1 cos x 0 0 I h( x).dx 3 2 sin x.dx I 2 2 0 1 cos x 2 cos 2 x.(cos 4 x sin 4 x).dx I 4 sin 4 x.dx I 0 0 2 4. sin x.dx I I 1 cos 2 x .dx cos x) 3 0 (sin x 0 3 dx I 4 sin x. cos x 6
1 x 3 .dx I x2 1 x 2 sin x 7 cos x 6 0 I .dx 2 dx 4 sin x 3 cos x 5 I 0 x. x 2 1 2 2 3 x. cos 2 x.dx I 1 dx 0 I x2 1 1x 1 2 4 (2 x 1). cos 2 x.dx I dx I 0 x. x 2 9 7 1 3 ( x sin x).dx x 3 . 1 x 2 .dx I I cos 2 x 0 0 2 dx I x. x 3 1 1 1 ( x 2 1).dx 1 2a I x15 . 1 3x 8 .dx; B x 2 .dx(a A x. 2a 0) x1 0 0 0 7 3 x .dx a 4 dx I x2. a2 x 2 .dx; B A (a 0) 1 x2 3 0 x(1 x) 0 1 1 x.dx 0 2 dx dx I A ;B 2x 1 0 x2 ( x 1)( x 2) x1 1 1 1 0 1 x 2 .dx dx A ;B x2 x4 x2 1 dx 1 1 I 2 2x e 3 0 2 22 dx 2 A ;B xx 1.dx 1 dx x. x 2 1 I 1 0 2x ex e 7 0 1 2 ln 3 x dx dx dx A ;B I 3 3 4 2x 1 x 1 ex 1 0 0 0 3 3 2 dx ( x 1 2)dx A ; (*)B x.e 2 x .dx I x2 8x 1 x 2x 1x1 0 0 0 x 1 dx (*) A ; 3 2 2 x 1x 1 e sin x . sin x. cos 3 x.dx I 1 0 1 e x dx 1 0 I 2 2 A 4 x dx; B x 2 x 2 .dx x 0e 1 0 1 ln 2 e 2 x .dx 2 1 x2 1 1 x2 I A dx; B .dx ex 1 x2 x 0 1 1 2 x 2 x.e 2 .dx I 0
1 dx I x e 1 6 4 sin xdx cos xdx 0 A B e 2 3 ln x. 2 ln x sin x cos x sin x cos x I .dx 0 0 x 0 1 e x .dx 4 e (1 e x ) 2 .dx cos 2 x. cos 2 x.dx A B I x ex 0e e2x 1 0 0 ln 2 (1 e x )dx cos 2 xdx 6 I A ex 1 sin 2 x 0 0 ln 2 5.dx I ex 5 0 2 2 x2 A x 1.dx; B 2 x 3 .dx sin 2 x. cos 3 x; y 0 0 y 0 va x 0; x 2 1 2 I 2x 1 x .dx; 1 ex; y x y e va x 1 5 3 I x 3 .dx x41 3x 12 x 5 1 2 sin 2 y ;y 1 va x 0; x 5 2 2 2 2 I x 4x 3 x 4 x .dx 0 x2 y 2 x; y 3x 2 3 1 2 3 2 A x 2 .dx; B x 4x 4 x .dx; x2 x2; y2 1 0 y x 2 x2 y 4x 3; y 3x 3 8 I cot gx tgx .dx; x2 8 x2; y y va y 8 8 x cos 3 x. sin 3 x sin 3 x. cos 3 x .dx; I x 2 1; y 0 y x 5 cos 3x. cos 3 x sin 3x. sin 3 x .dx; I 4 2 x2 ( P) : y 4x 3 I 1 sin x .dx; 0 3 x3 2x 2 I x .dx; ( x 1) 5 x; y e x va x 1 y 0
x2; y D y x sin 3 x; y cos 3 x va truc Oy voi 0 y x 4 x3 2x 2 y 0; (C) : y 4x 3 1 cos 4 x sin 4 x ; x D y 0; y ;x 2 4x y 4 x 1 ≤x≤1) ( x 4) 2 y2 2 (E) : 1 (C ) : y x 4 16 12 (C ) : y .x 2 x2 1 2 D y ;y x2 1 2 x2 (C ) : y y2 (4 x) 3 ; y 2 ( P) : y 2 D 4x 2x 2 y 2 va (P2 ) : x 1 3 y 2 ( P1 ) : x x.( x 1) 2 (C ) : y D y tgx; x 0; x ;y 0 3 ≤x≤ S y x. ln x; y 0; x 1; x e x2 y2 (E) : 1 a2 b2
(tõ n¨m 2002 trë l¹i ) x2 y 4 x 3 va y x3 x2 x2 y 4 va y 4 42 23 dx I x x2 4 5 (1 2 sin 2 x)dx 4 I 1 sin 2 x 0 2 x2 I x .dx 0 2 x.dx I 1 x1 1 e 1 3 ln x . ln xdx I x 1 3 ln( x 2 I x).dx 2