高等数学第四章不定积分习题DOC

海量资源，欢迎共阅第四章不定积分§4–1不定积分的概念与性质一.填空题1．若在区间上F(x)f(x)，则F(x)叫做f(x)在该区间上的一个,f(x)的所有原函数叫做f(x)在该区间上的。2．F(x)是f(x)的一个原函数，则y=F(x)的图形为ƒ(x)的一条.因为d(arcsinx)11x2,所以arcsinx是的一个原函数。dx若曲线y=ƒ(x)上点(x,y)的切线斜率与x3成正比例，并且通过点A(1,6)和B(2,该曲线方程为。二．是非判断题若fx的某个原函数为常数，则fx0.[]xdx一切初等函数在其定义区间上都有原函数.[]ffxdx3．.[]若fx在某一区间内不连续，则在这个区间内fx必无原函数.[]ylnax与ylnx是同一函数的原函数.[]三．单项选择题1．c为任意常数，且F'(x)=f(x),下式成立的有。（A）F'(x)dxf(x)+c;（B）f(x)dx=F(x)+c;（C）F(x)dxF'(x)+c;(D)f'(x)dx=F(x)+c.2.F(x)和G(x)是函数f(x)的任意两个原函数，f(x)0，则下式成立的有。（A）F(x)=cG(x);（B）F(x)=G(x)+c;（C）F(x)+G(x)=c;(D)F(x)G(x)=c.3．下列各式中是f(x)sin|x|的原函数。(A)ycos|x|;(B)y=-|cosx|;1(c)y=cosx,x0,(D)y=cosxc,x0,、cc任意常数。cosx2,x0;cosxc2,x0.124.F(x)f(x),f(x)为可导函数，且f(0)=1,又F(x)xf(x)x2，则f(x)=.(A)2x1(B)x21(C)2x1(D)x21设f(sin2x)cos2x，则f(x)=.(A)sinx1sin2xc;(B)x1x2c;(C)sin2x1sin4xc;(D)x21x4c;2222设a是正数，函数f(x)ax,(x)axloga(A)f(x)是(x)的导数；(B)(x)是f(x)的导数；e,则.(C)f(x)是(x)的原函数；(D)(x)是f(x)的不定积分。四．计算题3．（x1)(x31)dx4.(1x)2dx3x5.ex(1ex)dx6.32xe3xdxxx27.x222x2dx8.4sin3x1dxsin2xsin)(cosxx222dx10.1cos2xdx1cos2x11.cos2xdx12.223x332xdxsin2xcos2x13.(323x)dx14.secx(secxtanx)dx1x21x24815.(11)xxdx16.x2五．应用题1xdx1x一曲线通过点(e2,3),且在任一点处的切线的斜率等于该点横坐标的倒数,求该曲线的方程.一物体由静止开始运动,经t秒后的速度是3t2(米/秒),问:(1)(2)一、填空题在3秒后物体离开出发点的距离是多少?物体走完360米需要多少时间§4-2换元积分法1.dxd(ax)((a0))2.dxd(7x3)xdx5.xdxd(x2)4.xdxd(5x2)d(1x2)6.x2dxd(23x3)7.()8.xxe2xdxde2xe2dxd(1e2)9.xe2x2dxd()10.cos(x31)dxd()dxd(5lnx)12.dxd(35lnx)xx13.sin(t)dtd()14.dx1x2d(1arcsinx)15．1dx1xx21x21d1(1)2x1(1)2xdxx16.若f(x)dxF(x)c,则f(axb)dx(a0)二．是非判断题1．lnxdx1d111c.[]xxx2x22．1xdx2arctg.[]xcx13．设，则farcsinx.[]1x2fxdxsinxcdxxc已知flnx1,0x1,且f00，且x,x0,sin2xdxx,1x,1.[]sin3xc3fxex1,0x.[]6．若fxdxFxc，则fgxdxFgxc.[]三．单项选择题1.f(3x)dx(A)1f(x)c;(B)133.f(3x)c;(C)3f(x)c;(D)3f(3x)c;2.f(x)1[f(x)]2dx.(A)ln|1f(x)|c;(B)1ln|1[f(x)]2|c;2arctan[f(x)]c;(D)1arctan[f(x)]c.23.1x2dx.x(A)12ln|x|xC(B)12ln|x|xCxx(C)12ln|x|C(D)ln|x|xCx32x23x4.dx..2x(A)3x2ln3(3)xc;(B)3223x2x()x1c2(C)23x323xc3cln3ln22ln3ln225.1x7x(1x7)dx.50(A)1ln|x7(B)|c;ln|x7|c;7(1x7)271x7（）C1ln|x6|c;(D)1ln|x6|c;6(1x6)261x66.|x|dx.(A)1|x|2c;(B)1x2c;(c)1x|x|c;(D)1x2c;22227.e3x1dx.ex111212e2xexxc;e2xexxc;212e2xexc;e2xexc.8.e1sin2xsin2x的全体原函数是.(A)e1sin2x;(B)e1sin2xc;(C)e1sin2xc(D)e1sin2xc四．计算题1．x(2x23)30dx2.43．e7xdx4.lnxdx1dx(12x)3x5.excos(ex)dx6.cosxesinxdx7.tg10xsec2xdx8.sin3xdx9.cosxsinxdx10.dxcosxsinxxsin2xa2x2(x21)323.x2dx24.1dx25.27.x29dx26.x12xdx1dx11x21ex28.dx4-3分部积分法一．单项选择题1.xf""(x)dx.(A)xf'(x)f(x)c;(B)xf'(x)f'(x)c;(c)xf'(x)f(x)c;(D)xf'(x)f(x)dx.sinxln(tanx)dx.－cosxln(tanx)+ln|tanx2|c;(B)cosxln(tanx)+ln|cscx－cotx|+c;(c)ln(tanx)+ln|tanx2|c;(D)－cosxln(tanx)+ln|sinx|+c.xsin2xdx.1x21xsin2xc;(B)1x21cos2xc;4448(C)xcosx－sinx+c;(D)14x21cos2xc;8arcsinxdx.x2(A)1arcsinxln|cscxcotx|c;(B)1arcsinxln|cotxcscx|c;xx(C)1arcsinxln|11x2|c;(D)1arcsinxln|11x2|c;xarctanexexxxxdx.(A)exarctanex1ln(1e2x)c;(B)1ln(1e2x)exarctanexxc;22(C)arctanex(ex1)c;(D)exarctanexx1ln(1e2x)c;26.(lnx)2dx.x(A)1(ln2x2lnx2)c;(B)ln2x2lnx1c;(C)1xxln2xlnxx1c;x(D)exarctanexxx1ln(1e2x2)c.7．(arcsinx)2dx.1x2(A)arcsinx(xarcsinx2)2xc;(B)arcsinx(xarcsinx+21x2)2xc;521x21、2、(C)arcsinx(xarcsinx+2二．计算题x2lnxdxx2cosxdx)c;(D)arcsinx(xarcsinx+22)c;3、xtg2xdx4、xcosxdxsin3x5、e3xdx6、(x22x5)exdx7、(lnx)2dx8、cos(lnx)dx9、(lnx)3x2dx10、xtgxsec4xdx4-4几种特殊类型的积分（一）一．单项选择题1．x4x45x24dx.x8arctanx1arctanxc;(B)x3231c;3arctanxlnx2(x22．41)c;xdx.x8c.3arctanxx42x21(A)1ln|x2(21)|c;1ln|x2(21)|c;42x2(21)42x2(21)(C)1ln|x221)|c;1ln|x12|c.42x2(21)42x2123．x3dx(A)x834331arctanx2c;(B)431arctanx4c323(C)1arctanx4c(D)1arctanx2c1x232334．dx.x(x102)(A)lndx+arctanx5c;(B)1ln(x102)c;1(x102)x10ln(2c1);x10ln(x52)c20x1026x105．3x2x22x5dx.(A)3ln|x22x5|1arctanx12;(B)3x2tanx1c;22222(C)32(x22x5)arctan2x12c;(D)ln|x22x5|tanx1c2二．计算题1、x2xx3x2x1dx2、2x3dxx23x103、x5x48dx4、x21dxx3x5、x(x1)(x2)(x3)(x1)2(x1)dx6、3dxx317、dxx(x21)8、dx(x21)(x2x)9、dxx4110、dx(x21)(x2x1)4-5几种特殊类型的积分（二）一．单项选择题1．1的全体原函数是———。1sinxtanx1c;(B)sinx2x1tan2c;(C)tanx1sinx若c;(D)tanx+1ccosxu211R(sin2x,cos2x)dxR(1u2,1u2)1u2du,则u(A)tanx(B)cotx22(C)tanx(D)cotx54sinxcosxdx.sin4xcoc4x1arctan(cos2x)c;2(C)arctan(cos2x)+c,(D)1ln|sin2x1|c.(B)1arctan(cos2x)c21cosxdx.1cosx(A)x+2cotx+cscx+c;(C)-x+2(cscx-cotx)+c;sin2x1(B)-x-2cotx+c;(D)-x+cscx-cotx+csinx(2cscxcotx1)dx.sin3x2xsinxcotxc(C)2sinxcotxc;(D)xcscxcotxc二．计算题(B)2xsinxcotxc1.dx2.2sinxdx2sinx2sinx3.1tgxdx4.1dxsin2x1sinxcosx5.sin5xdx6.1dxcos2xsin3xcosx7.1sin2xdx8.sinxdxsinxcosx9.1dx10.sinxcosxdx3sinx4cosxsinxcosx1(2x3)2`11.1dx12..x31dx13.13x1dxdx14.x1x4x1dx15.11x6(1x)5(131x)dx16.1xdx.1xx第四章自测题一.填空题(2x)1x1.dx.2.sin2x1sin2xdx.3.11.4.sinx3dxxxarctanxdx.5．已知ƒ'(x)=│x│,且f(2)a，则ƒ(x)=。二.单项选择题1.对于不定积分f(x)dx,下列等式中是正确的.(A)df(x)dxf(x)；(B)f(x)dxf(x)；(C)df(x)f(x)；(D)df(x)dxf(x)；dx函数f(x)在(,)上连续,则df(x)dx等于(A)f(x)；(B)f(x)dx；(C)f(x)c；(D)f(x)dx若F(x)和G(x)都是f(x)的原函数,则.(A)F(x)G(x)0；(B)F(x)G(x)0；.C)F(x)G(x)c,(常数)；(D)F(x)G(x)c,(常数)；三.计算下列各题1.xa2x2dx；2.x1x24x13dx；6．[f(x)]f(x)dx。1x2ex13.xarccosxdx；4.xexdx；5.xsin2xdx；6.ln(ex1)dx.x1x2exsinxcosxxsin2x7.dx8.dx56131x4x(1x)9.x3dx10.1dxxx21x(1x)11.1dx12.dxdxx2x113.x1dx14.xdx5xx21x015．设f(x)x10x1，求f(x)dx。2xx1四．设f(sin2x)cos2xtan2x,当0