高等数学课后习题及参考答案（第四章）习题411.求下列不定积分:1(1)dx;x2111解dxx2dxx21CC.x221x(2)xxdx;33112解xxdxx2dxx2Cx2xC.35121(3)dx;x11111解dxx2dxx2C2xC.1x12(4)x23xdx;77113解x23xdxx3dxx3Cx33xC.710131(5)dx;x2x5511131解dxx2dxx2CC.252xx1xx2(6)mxndx;nnmn11m解mxndxxmdxxmCxmC.nnm1m(7)5x3dx;5解5x3dx5x3dxx4C.4(8)(x23x2)dx;13解(x23x2)dxx2dx3xdx2dxx3x22xC.32dh(9)(g是常数);2gh11dh112h解h2dh2h2CC.2gh2g2gg(10)(x2)2dx;1解(x2)2dx(x24x4)dxx2dx4xdx4dxx32x24xC.3(11)(x21)2dx;12解(x21)2dx(x42x21)dxx4dx2x2dxdxx5x3xC.53(12)(x1)(x31)dx;13解(x1)(x31)dx(x2xx31)dxx2dxx2dxx2dxdx12325x3x2x2xC.335(1x)2(13)dx;x113135(1x)212xx242解dxdx(x22x2x2)dx2x2x2x2C.xx353x43x21(14)dx;x213x43x211解dx(3x2)dxx3arctanxC.x21x21x2(15)dx;1x2x2x2111解dxdx(1)dxxarctanxC.1x21x21x23(16)(2ex)dx;x31解(2ex)dx2exdx3dx2ex3ln|x|C.xx32(17)()dx;1x21x23211解()dx3dx2dx3arctanx2arcsinxC.1x21x21x21x2ex(18)ex(1)dx;x11ex解ex(1)dx(exx2)dxex2x2C.x(19)3xexdx;(3e)x3xex解3xexdx(3e)xdxCC.ln(3e)ln3123x52x(20)dx;3x2()x23x52x2352解dx[25()x]dx2x5C2x()xC.x32ln2ln333ln3(21)secx(secxtanx)dx;解secx(secxtanx)dx(sec2xsecxtanx)dxtanxsecxC.x(22)cos2dx;2x1cosx11解cos2dxdx(1cosx)dx(xsinx)C.22221(23)dx;1cos2x111解dxdxtanxC.1cos2x2cos2x2cos2x(24)dx;cosxsinxcos2xcos2xsin2x解dxdx(cosxsinx)dxsinxcosxC.cosxsinxcosxsinxcos2x(25)dx;cos2xsin2xcos2xcos2xsin2x11解dxdx()dxcotxtanxC.cos2xsin2xcos2xsin2xsin2xcos2x1(26)(1)xxdx;x2357114解1xxdx(x4x4)dxx44x4C.x272.一曲线通过点(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.解设该曲线的方程为yf(x),则由题意得1yf(x),x1所以ydxln|x|C.x又因为