微積分課件不定積分典型例題

1、x2 11計(jì)算 x4 1解法x4 1(x2 2x1)(x2 2x而 (x2 2x1) (x2 2x1) 2(x2 1) 所x2 2xx2 x2 2xx2 2xxdx 2dx 1dx(x 2)2 (x 2)2 12 1 d( 2xd 2x12( 2x1)2 ( 2x1)2 2 1 2(x2 2x(x2 2x1)(x2 2x1)(x2 2x2x1)arctan( 2x1)1x4 dx 1x2 2x x2 2x x 122arctan( 2x1) 1 arctanx2 122解法2 1d(x 1當(dāng)xdx1 x2dxx2 d(x 1 1 arctan1(x 1 )2 x22222 ,x2 1 2 ,x

2、2 22 arctan x2 1 x22 x2 22 x4 dx x x2 1 22x2.x3 (x例3.(x4 1)2(x4 x2)解 令 u x2dx (x4 1) (x4 x2(u2 1)(u2 1(u2 1) (u2 AuuCuD, 兩邊乘以 u, u2 u 0, A1. 兩邊乘以 u1, 再令 u 1, B 12兩 邊 乘 以 再 令 u,0ABCC1令u1,D1dx (x4 1) (x4 x21u(u21)(u2 1 1 u2(uu2 1lnu 1lnu11ln(u2 1)1arctanu1ln x2 1ln(x2 1)1ln(x4 1)1arctanx2 1 1arctanx2

3、(x2 1)2(x4 x8 4 (x8 1)2dx (x8 1)2 x dx 8(x8 1)2 1 8x8 (x81)2d(x81ln(x81)88(x8 例5.求1 cos x1 cos x sin x解 令 tan x t,則1 cos1 1t1t1111t2 1tdx 2(1t2)(12 ( 1 tt1)dt t2 1lnt11ln(t2 1)arctant21ln(1sin x) x 2*6 x2 x4 x4 2u2 2 u2 21xx11 1 1 11 1 d( x1sin 1cos( cos2 xtan(x)xarcsin1 xx2 1x2 1t arcsint x12 求 (xa

4、)(bx)dx, 其中a(xa)(bx R2 xab)2其中R ba xu ab2 R2 u2duuRsint R2cos2tdt R21cos2t 2 R2(t 1sin2t)c R2 t R2 sintcost 1(ba)2arcsin2x(a2x (a b2x (a b)(xa)(bx) cos3*13I cosxsin xdxJ cosxsin x解 I J (11sin2x)dx x1cos2x(cosxsinx)(11sinI J cosxsin(cos2xsin2x)(11sin(cosxsin(11sin2x)cosdx1sin2x1ln(sin2x1)1sin *例14計(jì)算I

5、 1I dx1 1x1dxdxJ 1 x 1 1 I J 3 arctan3(x1)1 1xx2 I J 1 x3dx 1 1 x dx1 x3dxln(x1)3ln(x 1)從而可解出I.(略15 arcsin 2 xdx arcsin 2 xd(x1 1 2 （xdx2 2 (1x)1 16解 令I(lǐng) xx2 xx2 xt2 txt, x 1,dxxf(x)dxxdf(x) xf(x)ff(x)dx sin x c f(x)fsinxc xcosxsin x代入*）dxcosxsin sin c 18求xf(x)dxcosx2sin xx33336 x6 xx3 3 x3 6 x3 x3 6

6、 218ln(16 x)2325xdx x15x5x5x x55x x5x x5x*x x 1xsint1arctanxln xtt21 x1 x2 x11x2tt211 1 1 1 例e2x 3ex ex26arcsin 1 xarcsinx1 27eex xdx eex (ex)dx cos2xdxd(1sinxcosx) 1 sin xcos x1sin xln1sin xcosx (x2 (x2 (x 3x1)e (x2 x)ee(x2 3x 2(x2 x)exd(x2 x)ex3(x2 x)ex2 3xdxxdx1 1 1 sin 2xa2 cos2 x b2 sin 2 xd(a2

7、cos2 d(a2cos2 xb2sin2 a2cos2 xb2sin2 b2 b2 2a2 cos2 xb2sin2 (a2 cos2 xb2 sin2 x)2a2 cos2 xb2sin2 1ln x dx(xln1ln (xlnx ( x ln x )2xc xln11x2 d(x xx4 dx x 1 dx (x1)2 xx1 11x arctan 2222x2 22arctan22(x x 當(dāng)x 0,*2 12 12 1(x2 1) (x2 1) 12 1 xx1 x1 x1 xx2 a3 dx(1 x2)2x2 a例x2 a xxx3a2 1 1 1cost dt 1cost co

8、s2 1cos2 1 1x1xtt(x(x7 (t)7 (t2 1 ln 21lnxc. 3 3(1xx2)3x1(x)2 2(2x3 1x3 1xt2)4x2 x22 x x2 (x2)2dx x2x2(x ) dx x 2 x(2 *(1x7)dx x10(2 (1 x7)x6dx lnx 22)7 x(1x7 x7(2ln x71 x2n1dx xn xn 1xn dx n (11 xn )d(x 1(xn ln xn 1)c. 2x3 44(x121251 xsin xdxx2 1cos2cos2 2（分項(xiàng)分部積分） xtan x 2f(lnff(lnf(lnf(ln*例52求dxdf(lnf(lnf(lnd(f(ln xf(lnf(ln*例53求 maxx3x2 f(xm