Practica2
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PRACTICA DIRIGIDA V 1. CALCULAR LAS SIGUIENTES INTEGRALES: A) 2x-3x2-x-2dx Solución: 2x-3x2-x-2dx=2x-3x-2(x+1)dx 2x-3x-2(x+1)=Ax-2+B(x+1) 2x-3x-2(x+1)=Ax+1+B(x-2)x-2(x+1) 2x-3=xA+B+(A-2B) A+B=2 A-2B=-3 A=13, B=53 2x-3x2-x-2dx=13dx(x-2)+53dx(x+1) 2x-3x2-x-2dx=13lnx-2+53lnx+1+C B) xx+1(2x+1)dx Solución: xx+1(2x+1)=Ax+1+B(2x+1) xx+1(2x+1)=A2x+1+B(x+1)x+1(2x+1) x=x2A+B+(A+B) 2A+B=1 A+B=0 A=1,B=-1 xx+1(2x+1)dx=1dxx+1-1dx(2x+1) xx+1(2x+1)dx=lnx+1-12ln2x+1+C C) 2x+7x2-3x-10dx Solución: 2x+7x2-3x-10dx=2x+7x-5(x+2)dx 2x+7x-5(x+2)=A(x-5)+B(x+2) 2x+7=Ax+2+B(x-5) 2x+7=xA+B+(2A-5B) A+B=2 2A-5B=7 A=177, B=-37 2x+7x2-3x-10dx=177dx(x-5)-37dx(x+2) 2x+7x2-3x-10dx=177lnx+5-37lnx+2+C D) x5+x4-8x3-4xdx Solución: x5+x4-8x3-4xdx=x2+x+4+4x2+16x-8x3-4xdx x5+x4-8x3-4xdx=x2+x+4dx+4x2+16x-8x3-4xdx x5+x4-8x3-4xdx=x2+x+4dx+4x2+16x-8xx+2(x-2)dx…………(1) 4x2+16x-8xx+2(x-2)=Ax+B(x+2)+C(x-2) 4x2+16x-8=Ax2-4+Bx2-2x+C(x2+2x) 4x2+16x-8=x2A+B+C+x-2B+2C+(-4A) A+B+C=4 -2B+2C=16
-4A=-8 A=2, B=-3, C=5 x5+x4-8x3-4xdx=x2+x+4dx+4x2+16x-8xx+2(x-2)dx…………(1) x5+x4-8x3-4xdx=x2+x+4dx+2dxx-3dx(x+2)+5dx(x-2) x5+x4-8x3-4xdx=x55+x22+4x+2lnx-3lnx+2+5lnx-2+C E) xx-1x+3(x+5)dx Solución: xx-1x+3(x+5)=A(x-1)+B(x+3)+C(x+5) x=Ax2+8x+15+Bx2+4x-5+C(x2+2x-3) x=x2A+B+C+x8A+4B+2C+(15A-5B-3C) A+B+C=0 8A+4B+2C=1 15A-5B-3C=0 A=124, B=38, C=-512 xx-1x+3(x+5)dx=124dx(x-1)+38dx(x+3)-512dx(x+5) xx-1x+3(x+5)dx=124lnx-1+38lnx+3-512lnx+5+C F) 2x2+41x-91x-1x+3(x-4)dx Solución: 2x2+41x-91x-1x+3(x+5)=A(x-1)+B(x+3)+C(x-4) 2x2+41x-91=Ax2-x-12+Bx2-5x+4+C(x2+2x-3) 2x2+41x-91=x2A+B+C+x-A-5B+2C+(-12A+4B-3C) A+B+C=2 -A-5B+2C=41 -12A+4B-3C=-91 A=4, B=-7, C=5 2x2+41x-91x-1x+3(x-4)dx=4dx(x-1)-7dx(x+3)+5dx(x-4) 2x2+41x-91x-1x+3(x-4)dx=4lnx+1-7lnx+3+5lnx-4+C G) dxx2-1x2-5x+6 Solución: dxx2-1x2-5x+6=dxx-1x+1x-3(x-2) 1x-1x+1x-3(x-2)=A(x-1)+B(x+1)+C(x-3)+D(x-2) 1=Ax3-4x2+x+6+Bx3-6x2+11x-6+Cx3-2x2-x+2+D(x3-3x2-x+3) 1=x3A+B+C+D+x2-4A-6B-2C-3D+xA+11B-C-D+(6A-6B+2C+3D) A+B+C+D=0 -4A-6B-2C-3D=0 A+11B-C-D=0 6A-6B+2C+3D=1 A=14, B=-124, C=18, D=-13 dxx2-1x2-5x+6=14dx(x-1)-124dx(x+1)+18dx(x-3)-13dx(x-2) dxx2-1x2-5x+6=14lnx-1-124lnx+1+18lnx-3-13lnx-2+C H) 32x2x-14x2-16x+15dx Solución: 32x2x-14x2-16x+15dx=32x2x-12x-3(2x-5)dx
32x2x-12x-3(2x-5)=A(2x-1)+B(2x-3)+C(2x-5) 32x=A4x2-16x+15+B4x2-12x+5+C(4x2-8x+3) 32x=x24A+4B+4C+x-16A-12B-8C+(15A+5B+3C) 4A+4B+4C=0 -16A-12B-8C=32 15A+5B+3C=0 A=2, B=-12, C=10 32x2x-14x2-16x+15dx=2dx(2x-1)-12dx(2x-3)+10dx(2x-5) 32x2x-14x2-16x+15dx=ln2x-1-6ln2x-3+5ln2x-5+C I) 4x2-x-8x3-x2-2xdx Solución: 4x2-x-8x3-x2-2xdx=4x2-x-8xx-2(x+1)dx 4x2-x-8xx-2(x+1)=Ax+B(x-2)+C(x+1) 4x2-x-8=Ax2-x-2+Bx2+x+C(x2-2x) 4x2-x-8=x2A+B+C+x-A+B-2C+(-2A) A+B+C=4 -A+B-2C=-1 -2A=-8 A=4, B=1, C=-1 4x2-x-8x3-x2-2xdx=4dxx+1dx(x-2)-1dx(x+1) 4x2-x-8x3-x2-2xdx=4lnx+lnx-2-lnx+1+C J) 2x2-5x4-5x2+6dx Solución: 2x2-5x4-5x2+6dx=2x2-5x+3x-3x+2(x-2)dx 2x2-5x+3x-3x+2(x-2)=A(x+3)+B(x-3)+C(x+2)+D(x-2) 2x2-5=Ax3-3x2-2x+23+Bx3+3x2-2x-23+Cx3-2x2-3x+32+D(x3+2x23x-32) 2x2-5=x3A+B+C+D+x2-3A+3B-2C+2D+x-2A-2B-3C-3D+(23A23B+32C-32D) A+B+C+D=0 -3A+3B-2C+2D=2 -2A-2B-3C-3D=0 23A-23B+32C-32D=-5 A=-123, B=123, C=-122, D=122 2x2-5x4-5x2+6dx=-123dx(x+3)+123dx(x-3)-122dx(x+2)+122dx(x-2) 2x2-5x4-5x2+6dx=-123lnx+3+123lnx-3-122lnx+2+122lnx-2+C
2. CALCULAR LAS SIGUIENTES INTEGRALES: A) 2x-1(x-1)3dx Solución: 2x-1(x-1)3=A(x-1)+B(x-1)2+C(x-1)3 2x-1=Ax2-2x+1+Bx-1+C 2x-1=x2A+x-2A+B+(A-B+C)
A=0 -2A+B=2 A-B+C=-1 A=0, B=2, C=1 2x-1(x-1)3dx=0dx(x-1)+2dx(x-1)2+1dx(x-1)3 2x-1(x-1)3dx=-2(x-1)-12(x-1)2+C B) x2-3x+2x(x2+2x+1)dx Solución: x2-3x+2x(x2+2x+1)dx=x2-3x+2x(x+1)2dx x2-3x+2x(x+1)2=Ax+B(x+1)+C(x+1)2 x2-3x+2=Ax2+2x+1+Bx2+x+Cx x2-3x+2=x2A+B+x2A+B+C+(A) A+B=1 2A+B+C=-3 A=2 A=2, B=-1, C=-6 x2-3x+2x(x2+2x+1)dx=2dxx-1dx(x+1)-6dx(x+1)2 x2-3x+2x(x2+2x+1)dx=2lnx-lnx+1+6(x+1)+C C) x2x3+5x2+8x+4dx Solución: x2x3+5x2+8x+4dx=x2x+1(x+2)2dx x2x+1(x+2)2=A(x+1)+B(x+2)+C(x+2)2 x2=Ax2+4x+4+Bx2+3x+2+C(x+1) x2=x2A+B+x4A+3B+C+(4A+2B+C) A+B=1 4A+3B+C=0 4A+2B+C=0 A=1, B=0, C=-4 x2x3+5x2+8x+4dx=1dx(x+1)+0dx(x+2)-4dx(x+2)2 x2x3+5x2+8x+4dx=lnx+1+4(x+2)+C D) 3x4-x2+1(x-1)5dx Solución: 3x4-x2+1(x-1)5=A(x-1)+B(x-1)2+C(x-1)3+D(x-1)4+E(x-1)5 3x4-x2+1=Ax4-4x3+6x2-4x+1+Bx3-3x2+3x-1+Cx2-2x+1+Dx-1+E 3x4-x2+1=x4A+x3-4A+B+x26A-3B+C+x-4A+3B-2C+D+(A-B+CD+E) A=3 -4A+B=0 6A-3B+C=-1 -4A+3B-2C+D=0 A-B+C-D+E=1 A=3, B=12, C=17, D=10, E=3 3x4-x2+1(x-1)5dx=3dx(x-1)+12dx(x-1)2+17dx(x-1)3+10dx(x1)4+3dx(x-1)5
3x4-x2+1(x-1)5dx=3lnx-1-12(x-1)-172(x-1)2-103(x-1)3-34(x-1)4+C E) x3-6x2+11x-5(x-2)4dx Solución: x3-6x2+11x-5(x-2)4=A(x-2)+B(x-2)2+C(x-2)3+D(x-2)4 x3-6x2+11x-5=Ax3-6x2+12x-8+Bx2-4x+4+Cx-2+D x3-6x2+11x-5=x3A+x2-6A+B+x12A-4B+C+(-8A+4B-2C+D) A=1 -6A+B=-6 12A-4B+C=11 -8A+4B-2C+D=-5 A=1, B=0, C=-1, D=1 x3-6x2+11x-5(x-2)4dx=1dx(x-2)+0dx(x-2)2-1dx(x-2)3+1dx(x-2)4 x3-6x2+11x-5(x-2)4dx=lnx-2+12(x-2)2-13(x-2)3+C F)
x2+1x3+9x2+27x+27dx Solución: x2+1x3+9x2+27x+27dx=x2+1(x-3)2dx x2+1(x-3)2=A(x-3)+B(x-3)2+C(x-3)3 x2+1=Ax2-6x+9+Bx-3+C x2+1=x2A+x-6A+B+(9A-3B+C) A=1 -6A+B=0 9A-3B+C=1 A=1, B=6, C=10 x2+1x3+9x2+27x+27dx=1dx(x-3)+6dx(x-3)2+10dx(x-3)3 x2+1x3+9x2+27x+27dx=lnx-3-6(x-3)-102(x-3)2+C
G) dxx4-x2 Solución: dxx4-x2=dxx+1(x-1)x2 1x+1(x-1)x2=A(x+1)+Bx-1+Cx+Dx2 1=Ax3-x2+Bx3+x2+Cx3-x+D(x2-1) 1=x3A+B+C+x2-A+B+D+x-C+(-D) A+B+C=0 -A+B+D=0 -C=0 -D=1 A=-12, B=12, C=0, D=-1 dxx4-x2=-12dx(x+1)+12dx(x-1)+0dxx-1dxx2 dxx4-x2=-12lnx+1+12lnx-1+1x+C H) x2(x+2)2(x+4)2dx Solución: x2(x+2)2(x+4)2=Ax+2+B(x+2)2+C(x+4)+D(x+4)2 x2=Ax3+10x2+32x+32+Bx28x+16+Cx3+8x2+20x+16+D(x2+4x+4)
x2=x3A+C+x210A+B+8C+D+x32A8B+20C+4D+(32A+16B+16C+4D) A+C=0 10A+B+8C+D=1 32A-8B+20C+4D=0 32A+16B+16C+4D=0 A=-76, B=19, C=76, D=209 x2(x+2)2(x+4)2dx=76dx(x+2)+19dx(x+2)2+76dx(x+4)+209dx(x+4)2 x2(x+2)2(x+4)2dx=-76lnx+2-19(x+2)+76lnx+4-209(x+4)+C I)
(x-1)(x3+2x2+x)dx Solución: (x-1)(x3+2x2+x)dx=(x-1)x(x+1)2dx (x-1)x(x+1)2=Ax+B(x+1)+C(x+1)2 x-1=Ax2+2x+1+Bx2+x+Cx x-1=x2A+B+x2A+B+C+(A) A+B=0 2A+B+C=1 A=-1 A=-1, B=1, C=2 (x-1)(x3+2x2+x)dx=-1dxx+1dx(x+1)+2dx(x+1)2 (x-1)(x3+2x2+x)dx=-lnx+lnx+1-2(x+1)+C
J)
x3-2x2+4x3(x-2)2dx Solución: x3-2x2+4x3(x-2)2=Ax+Bx2+Cx3+D(x-2)+E(x-2)2 x3-2x2+4=Ax4-4x3+4x2+Bx3-4x2+4x+Cx2-4x+4+Dx4-2x3+E(x3) x3-2x2+4=x4A+D+x3-4A+B-2D+E+x24A-4B+C+x4B-4C+(4C) A+D=0 -4A+B-2D+E=1 4A-4B+C=-2 4B-4C=0 4C=4 A=14, B=1, C=1, D=-14, E=12 x3-2x2+4x3(x-2)2dx=14dxx+1dxx2+1dxx3-14dx(x-2)+12dx(x-2)2 x3-2x2+4x3(x-2)2dx=14lnx-1x-12x2-14lnx-2-12(x-2)+C
3. CALCULAR LAS SIGUIENTES INTEGRALES: A) 2x2-3x-3x-1(x2-2x+5)dx Solución: 2x2-3x-3x-1(x2-2x+5)=A(x-1)+Bx+C(x2+x+1) 2x2-3x-3=Ax2+x+1+Bx2-x+C(x-1) 2x2-3x-3=x2A+B+xA-B+C+(A-C) A+B=2 A-B+C=-3
A-C=-3 A=-43, B=103, C=53 2x2-3x-3x-1(x2-2x+5)=-43dx(x-1)+532x+1(x2+x+1)dx 2x2-3x-3x-1(x2-2x+5)=-43lnx+1+53lnx2+x+1+C B) 3x-4x-2(x2+3x-1)dx Solución: 3x-4x-2(x2+3x-1)=A(x-2)+Bx+C(x2+3x-1) 3x-4=Ax2+3x-1+Bx2-2x+C(x-2) 3x-4=x2A+B+x3A-2B+C+(-A-2C) A+B=0 3A-2B+C=3 -A-2C=-4 A=29, B=-29, C=179 3x-4x-2(x2+3x-1)dx=29dx(x-2)-192x-17(x2+3x-1)dx 3x-4x-2(x2+3x-1)dx=29lnx-2-192x+3-20(x2+3x-1)dx 3x-4x-2(x2+3x-1)dx=29lnx-2-192x+3x2+3x-1dx+209dx(x2+3x-1) 3x-4x-2(x2+3x-1)dx=29lnx-2-19lnx2+3x-1+209dx(x+32)2-(134)2 3x-4x-2(x2+3x-1)dx=29lnx-2-19lnx2+3x-1+20912413lnx+32134x+32+134+C 3x-4x-2(x2+3x-1)dx=29lnx-2-19lnx2+3x-1+20913ln2x+3132x+3+13+C C) x2(1-x4)dx Solución: x2(1-x4)dx=-x2x+1x-1(x2+1)dx x2x+1x-1(x2+1)=A(x+1)+B(x-1)+Cx+D(x2+1) x2=Ax3-x2+x-1+Bx3+x2+x+1+Cx3-x+D(x2-1) x2=x3A+B+C+x2-A+B+D+xA+B-C+(-A+B-D) A+B+C=0 -A+B+D=1 A+B-C=0 -A+B-D=0 A=-14, B=14, C=0, D=12 x2(1-x4)dx=--14dx(x+1)+14dx(x-1)+12dxx2+1 x2(1-x4)dx=14dx(x+1)-14dx(x-1)-12dxx2+1 x2(1-x4)dx=14lnx+1-14lnx-1-1211arctgx1+C x2(1-x4)dx=14lnx+1-14lnx-1-12arctgx+C D) xx6-1dx Solución: xx6-1dx=xx+1x2-x+1x-1(x2+x+1)dx xx+1x2-x+1x-1(x2+x+1)=A(x+1)+Bx+Cx2-x+1+Dx1+Ex+F(x2+x+1) x=Ax5-x4+x3-x2+x-1+Bx5+x4-x2-x+Cx4+x3-x1+Dx5+x4+x3+x2+x+1+Ex5-x4+x2-x+F(x4-x3+x-1)
x=x5A+B+D+E+x4-A+B+C+D-E+F+x3A+C+D-F+x2-A-B+D+E+xAB-C+D-E+F+(-A-C+D-F) A+B+D+E=0 -A+B+C+D-E+F=0 A+C+D-F=0 -A-B+D+E=0 A-B-C+D-E+F=1 -A-C+D-F=0 A=16, B=-16, C=-16, D=16, E=-16, F=16 xx6-1dx=16dx(x+1)-16x+1x2-x+1dx+16dx(x-1)-16x-1(x2+x+1)dx xx6-1dx=16lnx+1-1122x+2x2-x+1dx+16lnx-1-1122x-2(x2+x+1)dx xx6-1dx=16lnx+1-1122x-1+3x2-x+1dx+16lnx-1-1122x+13(x2+x+1)dx xx6-1dx=16lnx+1-1122x-1x2-x+1dx-14dxx2-x+1+16lnx-11122x+1(x2+x+1)dx+14dx(x2+x+1) xx6-1dx=16lnx+1-112lnx2-x+1-123arctg2x-13+16lnx-1112lnx2+x+1+123arctg(2x+13) E) dxx+12(x2+1)dx Solución: 1x+12(x2+1)=A(X+1)+Bx+12+Cx+D(x2+1) 1=Ax3+x2+x+1+Bx2+1+Cx3+2x2+x+D(x2+2x+1) 1=x3A+C+x2A+B+2C+D+xA+C+2D+(A+B+D) A+C=0 A+B+2C+D=0 A+C+2D=0 A+B+D=1 A=12, B=12, C=-12, D=0 dxx+12(x2+1)dx=12dx(x+1)+12dx(x+1)2-12xx2+1dx dxx+12(x2+1)dx=12lnx+1-12(x+1)-12122xx2+1dx dxx+12(x2+1)dx=12lnx+1-12(x+1)-14lnx2+1+C F) x5+2x3+4x+4x4+2x3+2x2dx Solución: x5+2x3+4x+4x4+2x3+2x2dx=x-2+4x3+4x2+4x+4x4+2x3+2x2dx x5+2x3+4x+4x4+2x3+2x2dx=(x2)dx+4x3+4x2+4x+4x2(x2+2x+2)dx 4x3+4x2+4x+4x2(x2+2x+2)=Ax+Bx2+Cx+Dx2+2x+2 4x3+4x2+4x+4=Ax3+2x2+2x+Bx2+2x+2+Cx3+D(x2) 4x3+4x2+4x+4=x3A+C+x22A+B+D+x2A+2B+(2B) A+C=4 2A+B+D=4 2A+2B=4 2B=4 A=2, B=2, C=2, D=-2 x5+2x3+4x+4x4+2x3+2x2dx=(x2)dx+4x3+4x2+4x+4x2(x2+2x+2)dx
x5+2x3+4x+4x4+2x3+2x2dx=x22-2x+2dxx+2dxx2+2x2x2+2x+2dx x5+2x3+4x+4x4+2x3+2x2dx=x22-2x+2lnx-2x+2x+2-4x2+2x+2dx x5+2x3+4x+4x4+2x3+2x2dx=x22-2x+2lnx-2x+2x+2x2+2x+2dx4dxx2+2x+2 x5+2x3+4x+4x4+2x3+2x2dx=x22-2x+2lnx-2x+lnx2+2x+24dx(x+1)2+(1)2 x5+2x3+4x+4x4+2x3+2x2dx=x22-2x+2lnx-2x+lnx2+2x+24arctgx+1+C G) 3x2+3x+1x3+2x2+2x+1dx Solución: 3x2+3x+1x3+2x2+2x+1dx=3x2+3x+1x+1(x2+x+1)dx 3x2+3x+1x+1(x2+x+1)=A(x+1)+Bx+C(x2+x+1) 3x2+3x+1=Ax2+x+1+Bx2+x+C(x+1) 3x2+3x+1=x2A+B+xA+B+C+(A+C) A+B=3 A+B+C=3 A+C=1 A=1, B=2, C=0 3x2+3x+1x3+2x2+2x+1dx=1dx(x+1)+2x(x2+x+1)dx 3x2+3x+1x3+2x2+2x+1dx=lnx+1+2x+1-1(x2+x+1)dx 3x2+3x+1x3+2x2+2x+1dx=lnx+1+2x+1(x2+x+1)dx-dx(x2+x+1) 3x2+3x+1x3+2x2+2x+1dx=lnx+1+lnx2+x+1-dx(x+12)2+(34)2 3x2+3x+1x3+2x2+2x+1dx=lnx+1+lnx2+x+1-23arctg2x+13+C H) 2x2+x+1x2+3(2x2+x+5)dx Solución: 2x2+x+1x2+3(2x2+x+5)=Ax+Bx2+3+Cx+D(2x2+x+5) 2x2+x+1=A2x3+x2+5x+B2x2+x+5+Cx3+3x+D(x2+3) 2x2+x+1=x32A+C+x2A+2B+D+x5A+B+3C+(5B+3D) 2A+C=0 A+2B+D=2 5A+B+3C=1 5B+3D=1 A=1, B=2, C=-2, D=-3 2x2+x+1x2+3(2x2+x+5)dx=x+2(x2+3)dx-2x+32x2+x+5dx 2x2+x+1x2+3(2x2+x+5)dx=122x+4x2+3dx-124x+62x2+x+5dx 2x2+x+1x2+3(2x2+x+5)dx=122xx2+3dx+2dxx2+3124x+1+52x2+x+5dx 2x2+x+1x2+3(2x2+x+5)dx=12lnx2+3+213arctgx3124x+12x2+x+5dx-53dx2x2+x+5 2x2+x+1x2+3(2x2+x+5)dx=12lnx2+3+213arctgx3-12ln2x2+x+552dx(2x+122)2+(398)2 2x2+x+1x2+3(2x2+x+5)dx=12lnx2+3+213arctgx3-12ln2x2+x+55239arctg4x+139+C
4) calcular las siguientes integrales
a)3x(x2+x+3)3dx 3xx2+x+33dx=3x(x+122+1122)3dx x+12=112tan∝ , derivando, dx=112 (sec∝)2d∝ 3(112tan∝-12)112 (sec∝)2d∝(114(sec∝)2)3 3114tan∝(sec∝)2d∝(114(sec∝)2)3-3114(sec∝)2d∝(114(sec∝)2)3 3tan∝sec∝d∝1214sec∝5b)1(X3-1)2dx 1(X3-1)2=AX-1+B(X-1)2+C2x+1+D(x2+x+1)+E2X+1+F(x2+x+1)2 1=Ax-1x2+x+12+B(x2+x+1)2+( C2x+1+D) X-12x2+x+1+( E2X+1+F) (X1)2 RESOLVIENDO A=-29,B=19, C=19, D=29, E=16, F=16 (-29X-1+19(X-1)2+192x+1+29(x2+x+1)+162X+1+16(x2+x+1)2)dx =-29.lnx-1-19.1x-1+19lnx2+x+1+29.1(x+12)2+(32)216.1x2+x+1+16.1((x+12)2+(32)2)2dx =-29.lnx-1-19.1x-1+19lnx2+x+1+29.233tan-1(23x+33)16.1x2+x+1+16.(439tan-123x+33+2x+13x2+3x+3) 1(X3-1)2dx==-29.lnx-1-19.1x-1+19lnx2+x+1+4327tan-1(23x+33)16.1x2+x+1+2327tan-123x+33+118.2x+1x2+x+1
1(X3-1)2dx=-29.lnx-1-19.1x-1+19lnx2+x+1+6327tan-1(23x+33)+118.2x2x2+x+1 c)5x2-12(x2-6x+13)2dx 5x2-12(x2-6x+13)2=A2x-6+Bx2-6x+13+C2x-6+D(x2-6x+13)2 5x2-12=A2x-6+Bx2-6x+13+(C2x-6+D) A=0, B=5, C=15, D=13 5x2-6x+13+152x-6+13(x2-6x+13)2dx 5(x-3)2+4dx-15x2-6x+13+13(x-32+4)2dx 5x2-12(x2-6x+13)2dx=52.tan-1(x-22)-15x2-6x+13+138(tan-1x-222+x-3x26x+13) 5x2-12(x2-6x+13)2dx=5316tan-1(x-22)+18.13x-159x2-6x+13+C
d)x3(x2+2x+2)2dx
x3(x2+2x+2)2=A2X+2+Bx2+2x+2+C2X+2+E(x2+2x+2)2 x3=(A2X+2+B)( x2+2x+2)+ C2X+2+E RESOLVIENDO A=12, B=-3, C=1, E=2 (122X+2-3(x+1)2+1+2X+2+2((x+1)2+1)2)dx=12lnx2+2x+2-3tan1x+1+tan-1x+1+xx2+2x+2 x3(x2+2x+2)2dx=12lnx2+2x+2-2tan-1x+1+xx2+2x+2 e)dx(1+x2)4 x=tanθ;dx=secθ2dθ reemplazando secθ2dθsecθ8=cos6θdθ resolviendo dicha integral se obtiene que cos6θdθ=516θ+12senθcosθ+316senθcosθ(2cosθ2-1)-16(cosθsenθ)3 dx(1+x2)4=516tan-1x+12.xx2+1+316.xx2+1211+x2-1-16(xx2+1)3 dx(1+x2)4=516tan-1x+15x5+40x3+33x48x6+144x4+144x2+48+c
f)x3+x-1(x2+2)2dx x3+x-1(x2+2)2=A2X+Bx2+2+C2X+D(x2+2)2 x3+x-1=x2+2(A2X+B)+(C2X+D) RESOLVIENDO A=12, B=0 C=-12, D=-1 122Xx2+2+-122X-1(x2+2)2dx=12lnx2+2+12.1x2+2-1(x2+2)2dx x3+x-1(x2+2)2dx=12lnx2+2+12.1x2+2-(x4x2+2+28tan-1(2x2) x3+x-1(x2+2)2dx=12lnx2+2+2-x4x2+2-28tan-1(2x2)+c g)3x+5(2x2+x+1)2dx h)x9(x4-1)2dx dividiendo y por el algoritmo de la división se obtiene x9(x4-1)2=x+2x5-x(x4-1)2 2x5-x(x4-1)2=Ax-1+B(X1)2+C(X+1)+D(X+1)2+E2X+FX2+1+G2X+H(X2+1)2 resolviendo se obtiene que A=18, B=316, C=38, D=-116, E=-38, F=0,G=18, H=0 (x+38x-1+116(X-1)2+38X+1+-116X+12+-382XX2+1+182XX2+12)dx x9(x4-1)2dx=x22+38lnx-1-116.1x-1+38lnx+1+116.1x+1-38.lnx2+118.1x2+1+c
j)1(x2+9)3dx
-4A=-8 A=2, B=-3, C=5 x5+x4-8x3-4xdx=x2+x+4dx+4x2+16x-8xx+2(x-2)dx…………(1) x5+x4-8x3-4xdx=x2+x+4dx+2dxx-3dx(x+2)+5dx(x-2) x5+x4-8x3-4xdx=x55+x22+4x+2lnx-3lnx+2+5lnx-2+C E) xx-1x+3(x+5)dx Solución: xx-1x+3(x+5)=A(x-1)+B(x+3)+C(x+5) x=Ax2+8x+15+Bx2+4x-5+C(x2+2x-3) x=x2A+B+C+x8A+4B+2C+(15A-5B-3C) A+B+C=0 8A+4B+2C=1 15A-5B-3C=0 A=124, B=38, C=-512 xx-1x+3(x+5)dx=124dx(x-1)+38dx(x+3)-512dx(x+5) xx-1x+3(x+5)dx=124lnx-1+38lnx+3-512lnx+5+C F) 2x2+41x-91x-1x+3(x-4)dx Solución: 2x2+41x-91x-1x+3(x+5)=A(x-1)+B(x+3)+C(x-4) 2x2+41x-91=Ax2-x-12+Bx2-5x+4+C(x2+2x-3) 2x2+41x-91=x2A+B+C+x-A-5B+2C+(-12A+4B-3C) A+B+C=2 -A-5B+2C=41 -12A+4B-3C=-91 A=4, B=-7, C=5 2x2+41x-91x-1x+3(x-4)dx=4dx(x-1)-7dx(x+3)+5dx(x-4) 2x2+41x-91x-1x+3(x-4)dx=4lnx+1-7lnx+3+5lnx-4+C G) dxx2-1x2-5x+6 Solución: dxx2-1x2-5x+6=dxx-1x+1x-3(x-2) 1x-1x+1x-3(x-2)=A(x-1)+B(x+1)+C(x-3)+D(x-2) 1=Ax3-4x2+x+6+Bx3-6x2+11x-6+Cx3-2x2-x+2+D(x3-3x2-x+3) 1=x3A+B+C+D+x2-4A-6B-2C-3D+xA+11B-C-D+(6A-6B+2C+3D) A+B+C+D=0 -4A-6B-2C-3D=0 A+11B-C-D=0 6A-6B+2C+3D=1 A=14, B=-124, C=18, D=-13 dxx2-1x2-5x+6=14dx(x-1)-124dx(x+1)+18dx(x-3)-13dx(x-2) dxx2-1x2-5x+6=14lnx-1-124lnx+1+18lnx-3-13lnx-2+C H) 32x2x-14x2-16x+15dx Solución: 32x2x-14x2-16x+15dx=32x2x-12x-3(2x-5)dx
32x2x-12x-3(2x-5)=A(2x-1)+B(2x-3)+C(2x-5) 32x=A4x2-16x+15+B4x2-12x+5+C(4x2-8x+3) 32x=x24A+4B+4C+x-16A-12B-8C+(15A+5B+3C) 4A+4B+4C=0 -16A-12B-8C=32 15A+5B+3C=0 A=2, B=-12, C=10 32x2x-14x2-16x+15dx=2dx(2x-1)-12dx(2x-3)+10dx(2x-5) 32x2x-14x2-16x+15dx=ln2x-1-6ln2x-3+5ln2x-5+C I) 4x2-x-8x3-x2-2xdx Solución: 4x2-x-8x3-x2-2xdx=4x2-x-8xx-2(x+1)dx 4x2-x-8xx-2(x+1)=Ax+B(x-2)+C(x+1) 4x2-x-8=Ax2-x-2+Bx2+x+C(x2-2x) 4x2-x-8=x2A+B+C+x-A+B-2C+(-2A) A+B+C=4 -A+B-2C=-1 -2A=-8 A=4, B=1, C=-1 4x2-x-8x3-x2-2xdx=4dxx+1dx(x-2)-1dx(x+1) 4x2-x-8x3-x2-2xdx=4lnx+lnx-2-lnx+1+C J) 2x2-5x4-5x2+6dx Solución: 2x2-5x4-5x2+6dx=2x2-5x+3x-3x+2(x-2)dx 2x2-5x+3x-3x+2(x-2)=A(x+3)+B(x-3)+C(x+2)+D(x-2) 2x2-5=Ax3-3x2-2x+23+Bx3+3x2-2x-23+Cx3-2x2-3x+32+D(x3+2x23x-32) 2x2-5=x3A+B+C+D+x2-3A+3B-2C+2D+x-2A-2B-3C-3D+(23A23B+32C-32D) A+B+C+D=0 -3A+3B-2C+2D=2 -2A-2B-3C-3D=0 23A-23B+32C-32D=-5 A=-123, B=123, C=-122, D=122 2x2-5x4-5x2+6dx=-123dx(x+3)+123dx(x-3)-122dx(x+2)+122dx(x-2) 2x2-5x4-5x2+6dx=-123lnx+3+123lnx-3-122lnx+2+122lnx-2+C
2. CALCULAR LAS SIGUIENTES INTEGRALES: A) 2x-1(x-1)3dx Solución: 2x-1(x-1)3=A(x-1)+B(x-1)2+C(x-1)3 2x-1=Ax2-2x+1+Bx-1+C 2x-1=x2A+x-2A+B+(A-B+C)
A=0 -2A+B=2 A-B+C=-1 A=0, B=2, C=1 2x-1(x-1)3dx=0dx(x-1)+2dx(x-1)2+1dx(x-1)3 2x-1(x-1)3dx=-2(x-1)-12(x-1)2+C B) x2-3x+2x(x2+2x+1)dx Solución: x2-3x+2x(x2+2x+1)dx=x2-3x+2x(x+1)2dx x2-3x+2x(x+1)2=Ax+B(x+1)+C(x+1)2 x2-3x+2=Ax2+2x+1+Bx2+x+Cx x2-3x+2=x2A+B+x2A+B+C+(A) A+B=1 2A+B+C=-3 A=2 A=2, B=-1, C=-6 x2-3x+2x(x2+2x+1)dx=2dxx-1dx(x+1)-6dx(x+1)2 x2-3x+2x(x2+2x+1)dx=2lnx-lnx+1+6(x+1)+C C) x2x3+5x2+8x+4dx Solución: x2x3+5x2+8x+4dx=x2x+1(x+2)2dx x2x+1(x+2)2=A(x+1)+B(x+2)+C(x+2)2 x2=Ax2+4x+4+Bx2+3x+2+C(x+1) x2=x2A+B+x4A+3B+C+(4A+2B+C) A+B=1 4A+3B+C=0 4A+2B+C=0 A=1, B=0, C=-4 x2x3+5x2+8x+4dx=1dx(x+1)+0dx(x+2)-4dx(x+2)2 x2x3+5x2+8x+4dx=lnx+1+4(x+2)+C D) 3x4-x2+1(x-1)5dx Solución: 3x4-x2+1(x-1)5=A(x-1)+B(x-1)2+C(x-1)3+D(x-1)4+E(x-1)5 3x4-x2+1=Ax4-4x3+6x2-4x+1+Bx3-3x2+3x-1+Cx2-2x+1+Dx-1+E 3x4-x2+1=x4A+x3-4A+B+x26A-3B+C+x-4A+3B-2C+D+(A-B+CD+E) A=3 -4A+B=0 6A-3B+C=-1 -4A+3B-2C+D=0 A-B+C-D+E=1 A=3, B=12, C=17, D=10, E=3 3x4-x2+1(x-1)5dx=3dx(x-1)+12dx(x-1)2+17dx(x-1)3+10dx(x1)4+3dx(x-1)5
3x4-x2+1(x-1)5dx=3lnx-1-12(x-1)-172(x-1)2-103(x-1)3-34(x-1)4+C E) x3-6x2+11x-5(x-2)4dx Solución: x3-6x2+11x-5(x-2)4=A(x-2)+B(x-2)2+C(x-2)3+D(x-2)4 x3-6x2+11x-5=Ax3-6x2+12x-8+Bx2-4x+4+Cx-2+D x3-6x2+11x-5=x3A+x2-6A+B+x12A-4B+C+(-8A+4B-2C+D) A=1 -6A+B=-6 12A-4B+C=11 -8A+4B-2C+D=-5 A=1, B=0, C=-1, D=1 x3-6x2+11x-5(x-2)4dx=1dx(x-2)+0dx(x-2)2-1dx(x-2)3+1dx(x-2)4 x3-6x2+11x-5(x-2)4dx=lnx-2+12(x-2)2-13(x-2)3+C F)
x2+1x3+9x2+27x+27dx Solución: x2+1x3+9x2+27x+27dx=x2+1(x-3)2dx x2+1(x-3)2=A(x-3)+B(x-3)2+C(x-3)3 x2+1=Ax2-6x+9+Bx-3+C x2+1=x2A+x-6A+B+(9A-3B+C) A=1 -6A+B=0 9A-3B+C=1 A=1, B=6, C=10 x2+1x3+9x2+27x+27dx=1dx(x-3)+6dx(x-3)2+10dx(x-3)3 x2+1x3+9x2+27x+27dx=lnx-3-6(x-3)-102(x-3)2+C
G) dxx4-x2 Solución: dxx4-x2=dxx+1(x-1)x2 1x+1(x-1)x2=A(x+1)+Bx-1+Cx+Dx2 1=Ax3-x2+Bx3+x2+Cx3-x+D(x2-1) 1=x3A+B+C+x2-A+B+D+x-C+(-D) A+B+C=0 -A+B+D=0 -C=0 -D=1 A=-12, B=12, C=0, D=-1 dxx4-x2=-12dx(x+1)+12dx(x-1)+0dxx-1dxx2 dxx4-x2=-12lnx+1+12lnx-1+1x+C H) x2(x+2)2(x+4)2dx Solución: x2(x+2)2(x+4)2=Ax+2+B(x+2)2+C(x+4)+D(x+4)2 x2=Ax3+10x2+32x+32+Bx28x+16+Cx3+8x2+20x+16+D(x2+4x+4)
x2=x3A+C+x210A+B+8C+D+x32A8B+20C+4D+(32A+16B+16C+4D) A+C=0 10A+B+8C+D=1 32A-8B+20C+4D=0 32A+16B+16C+4D=0 A=-76, B=19, C=76, D=209 x2(x+2)2(x+4)2dx=76dx(x+2)+19dx(x+2)2+76dx(x+4)+209dx(x+4)2 x2(x+2)2(x+4)2dx=-76lnx+2-19(x+2)+76lnx+4-209(x+4)+C I)
(x-1)(x3+2x2+x)dx Solución: (x-1)(x3+2x2+x)dx=(x-1)x(x+1)2dx (x-1)x(x+1)2=Ax+B(x+1)+C(x+1)2 x-1=Ax2+2x+1+Bx2+x+Cx x-1=x2A+B+x2A+B+C+(A) A+B=0 2A+B+C=1 A=-1 A=-1, B=1, C=2 (x-1)(x3+2x2+x)dx=-1dxx+1dx(x+1)+2dx(x+1)2 (x-1)(x3+2x2+x)dx=-lnx+lnx+1-2(x+1)+C
J)
x3-2x2+4x3(x-2)2dx Solución: x3-2x2+4x3(x-2)2=Ax+Bx2+Cx3+D(x-2)+E(x-2)2 x3-2x2+4=Ax4-4x3+4x2+Bx3-4x2+4x+Cx2-4x+4+Dx4-2x3+E(x3) x3-2x2+4=x4A+D+x3-4A+B-2D+E+x24A-4B+C+x4B-4C+(4C) A+D=0 -4A+B-2D+E=1 4A-4B+C=-2 4B-4C=0 4C=4 A=14, B=1, C=1, D=-14, E=12 x3-2x2+4x3(x-2)2dx=14dxx+1dxx2+1dxx3-14dx(x-2)+12dx(x-2)2 x3-2x2+4x3(x-2)2dx=14lnx-1x-12x2-14lnx-2-12(x-2)+C
3. CALCULAR LAS SIGUIENTES INTEGRALES: A) 2x2-3x-3x-1(x2-2x+5)dx Solución: 2x2-3x-3x-1(x2-2x+5)=A(x-1)+Bx+C(x2+x+1) 2x2-3x-3=Ax2+x+1+Bx2-x+C(x-1) 2x2-3x-3=x2A+B+xA-B+C+(A-C) A+B=2 A-B+C=-3
A-C=-3 A=-43, B=103, C=53 2x2-3x-3x-1(x2-2x+5)=-43dx(x-1)+532x+1(x2+x+1)dx 2x2-3x-3x-1(x2-2x+5)=-43lnx+1+53lnx2+x+1+C B) 3x-4x-2(x2+3x-1)dx Solución: 3x-4x-2(x2+3x-1)=A(x-2)+Bx+C(x2+3x-1) 3x-4=Ax2+3x-1+Bx2-2x+C(x-2) 3x-4=x2A+B+x3A-2B+C+(-A-2C) A+B=0 3A-2B+C=3 -A-2C=-4 A=29, B=-29, C=179 3x-4x-2(x2+3x-1)dx=29dx(x-2)-192x-17(x2+3x-1)dx 3x-4x-2(x2+3x-1)dx=29lnx-2-192x+3-20(x2+3x-1)dx 3x-4x-2(x2+3x-1)dx=29lnx-2-192x+3x2+3x-1dx+209dx(x2+3x-1) 3x-4x-2(x2+3x-1)dx=29lnx-2-19lnx2+3x-1+209dx(x+32)2-(134)2 3x-4x-2(x2+3x-1)dx=29lnx-2-19lnx2+3x-1+20912413lnx+32134x+32+134+C 3x-4x-2(x2+3x-1)dx=29lnx-2-19lnx2+3x-1+20913ln2x+3132x+3+13+C C) x2(1-x4)dx Solución: x2(1-x4)dx=-x2x+1x-1(x2+1)dx x2x+1x-1(x2+1)=A(x+1)+B(x-1)+Cx+D(x2+1) x2=Ax3-x2+x-1+Bx3+x2+x+1+Cx3-x+D(x2-1) x2=x3A+B+C+x2-A+B+D+xA+B-C+(-A+B-D) A+B+C=0 -A+B+D=1 A+B-C=0 -A+B-D=0 A=-14, B=14, C=0, D=12 x2(1-x4)dx=--14dx(x+1)+14dx(x-1)+12dxx2+1 x2(1-x4)dx=14dx(x+1)-14dx(x-1)-12dxx2+1 x2(1-x4)dx=14lnx+1-14lnx-1-1211arctgx1+C x2(1-x4)dx=14lnx+1-14lnx-1-12arctgx+C D) xx6-1dx Solución: xx6-1dx=xx+1x2-x+1x-1(x2+x+1)dx xx+1x2-x+1x-1(x2+x+1)=A(x+1)+Bx+Cx2-x+1+Dx1+Ex+F(x2+x+1) x=Ax5-x4+x3-x2+x-1+Bx5+x4-x2-x+Cx4+x3-x1+Dx5+x4+x3+x2+x+1+Ex5-x4+x2-x+F(x4-x3+x-1)
x=x5A+B+D+E+x4-A+B+C+D-E+F+x3A+C+D-F+x2-A-B+D+E+xAB-C+D-E+F+(-A-C+D-F) A+B+D+E=0 -A+B+C+D-E+F=0 A+C+D-F=0 -A-B+D+E=0 A-B-C+D-E+F=1 -A-C+D-F=0 A=16, B=-16, C=-16, D=16, E=-16, F=16 xx6-1dx=16dx(x+1)-16x+1x2-x+1dx+16dx(x-1)-16x-1(x2+x+1)dx xx6-1dx=16lnx+1-1122x+2x2-x+1dx+16lnx-1-1122x-2(x2+x+1)dx xx6-1dx=16lnx+1-1122x-1+3x2-x+1dx+16lnx-1-1122x+13(x2+x+1)dx xx6-1dx=16lnx+1-1122x-1x2-x+1dx-14dxx2-x+1+16lnx-11122x+1(x2+x+1)dx+14dx(x2+x+1) xx6-1dx=16lnx+1-112lnx2-x+1-123arctg2x-13+16lnx-1112lnx2+x+1+123arctg(2x+13) E) dxx+12(x2+1)dx Solución: 1x+12(x2+1)=A(X+1)+Bx+12+Cx+D(x2+1) 1=Ax3+x2+x+1+Bx2+1+Cx3+2x2+x+D(x2+2x+1) 1=x3A+C+x2A+B+2C+D+xA+C+2D+(A+B+D) A+C=0 A+B+2C+D=0 A+C+2D=0 A+B+D=1 A=12, B=12, C=-12, D=0 dxx+12(x2+1)dx=12dx(x+1)+12dx(x+1)2-12xx2+1dx dxx+12(x2+1)dx=12lnx+1-12(x+1)-12122xx2+1dx dxx+12(x2+1)dx=12lnx+1-12(x+1)-14lnx2+1+C F) x5+2x3+4x+4x4+2x3+2x2dx Solución: x5+2x3+4x+4x4+2x3+2x2dx=x-2+4x3+4x2+4x+4x4+2x3+2x2dx x5+2x3+4x+4x4+2x3+2x2dx=(x2)dx+4x3+4x2+4x+4x2(x2+2x+2)dx 4x3+4x2+4x+4x2(x2+2x+2)=Ax+Bx2+Cx+Dx2+2x+2 4x3+4x2+4x+4=Ax3+2x2+2x+Bx2+2x+2+Cx3+D(x2) 4x3+4x2+4x+4=x3A+C+x22A+B+D+x2A+2B+(2B) A+C=4 2A+B+D=4 2A+2B=4 2B=4 A=2, B=2, C=2, D=-2 x5+2x3+4x+4x4+2x3+2x2dx=(x2)dx+4x3+4x2+4x+4x2(x2+2x+2)dx
x5+2x3+4x+4x4+2x3+2x2dx=x22-2x+2dxx+2dxx2+2x2x2+2x+2dx x5+2x3+4x+4x4+2x3+2x2dx=x22-2x+2lnx-2x+2x+2-4x2+2x+2dx x5+2x3+4x+4x4+2x3+2x2dx=x22-2x+2lnx-2x+2x+2x2+2x+2dx4dxx2+2x+2 x5+2x3+4x+4x4+2x3+2x2dx=x22-2x+2lnx-2x+lnx2+2x+24dx(x+1)2+(1)2 x5+2x3+4x+4x4+2x3+2x2dx=x22-2x+2lnx-2x+lnx2+2x+24arctgx+1+C G) 3x2+3x+1x3+2x2+2x+1dx Solución: 3x2+3x+1x3+2x2+2x+1dx=3x2+3x+1x+1(x2+x+1)dx 3x2+3x+1x+1(x2+x+1)=A(x+1)+Bx+C(x2+x+1) 3x2+3x+1=Ax2+x+1+Bx2+x+C(x+1) 3x2+3x+1=x2A+B+xA+B+C+(A+C) A+B=3 A+B+C=3 A+C=1 A=1, B=2, C=0 3x2+3x+1x3+2x2+2x+1dx=1dx(x+1)+2x(x2+x+1)dx 3x2+3x+1x3+2x2+2x+1dx=lnx+1+2x+1-1(x2+x+1)dx 3x2+3x+1x3+2x2+2x+1dx=lnx+1+2x+1(x2+x+1)dx-dx(x2+x+1) 3x2+3x+1x3+2x2+2x+1dx=lnx+1+lnx2+x+1-dx(x+12)2+(34)2 3x2+3x+1x3+2x2+2x+1dx=lnx+1+lnx2+x+1-23arctg2x+13+C H) 2x2+x+1x2+3(2x2+x+5)dx Solución: 2x2+x+1x2+3(2x2+x+5)=Ax+Bx2+3+Cx+D(2x2+x+5) 2x2+x+1=A2x3+x2+5x+B2x2+x+5+Cx3+3x+D(x2+3) 2x2+x+1=x32A+C+x2A+2B+D+x5A+B+3C+(5B+3D) 2A+C=0 A+2B+D=2 5A+B+3C=1 5B+3D=1 A=1, B=2, C=-2, D=-3 2x2+x+1x2+3(2x2+x+5)dx=x+2(x2+3)dx-2x+32x2+x+5dx 2x2+x+1x2+3(2x2+x+5)dx=122x+4x2+3dx-124x+62x2+x+5dx 2x2+x+1x2+3(2x2+x+5)dx=122xx2+3dx+2dxx2+3124x+1+52x2+x+5dx 2x2+x+1x2+3(2x2+x+5)dx=12lnx2+3+213arctgx3124x+12x2+x+5dx-53dx2x2+x+5 2x2+x+1x2+3(2x2+x+5)dx=12lnx2+3+213arctgx3-12ln2x2+x+552dx(2x+122)2+(398)2 2x2+x+1x2+3(2x2+x+5)dx=12lnx2+3+213arctgx3-12ln2x2+x+55239arctg4x+139+C
4) calcular las siguientes integrales
a)3x(x2+x+3)3dx 3xx2+x+33dx=3x(x+122+1122)3dx x+12=112tan∝ , derivando, dx=112 (sec∝)2d∝ 3(112tan∝-12)112 (sec∝)2d∝(114(sec∝)2)3 3114tan∝(sec∝)2d∝(114(sec∝)2)3-3114(sec∝)2d∝(114(sec∝)2)3 3tan∝sec∝d∝1214sec∝5b)1(X3-1)2dx 1(X3-1)2=AX-1+B(X-1)2+C2x+1+D(x2+x+1)+E2X+1+F(x2+x+1)2 1=Ax-1x2+x+12+B(x2+x+1)2+( C2x+1+D) X-12x2+x+1+( E2X+1+F) (X1)2 RESOLVIENDO A=-29,B=19, C=19, D=29, E=16, F=16 (-29X-1+19(X-1)2+192x+1+29(x2+x+1)+162X+1+16(x2+x+1)2)dx =-29.lnx-1-19.1x-1+19lnx2+x+1+29.1(x+12)2+(32)216.1x2+x+1+16.1((x+12)2+(32)2)2dx =-29.lnx-1-19.1x-1+19lnx2+x+1+29.233tan-1(23x+33)16.1x2+x+1+16.(439tan-123x+33+2x+13x2+3x+3) 1(X3-1)2dx==-29.lnx-1-19.1x-1+19lnx2+x+1+4327tan-1(23x+33)16.1x2+x+1+2327tan-123x+33+118.2x+1x2+x+1
1(X3-1)2dx=-29.lnx-1-19.1x-1+19lnx2+x+1+6327tan-1(23x+33)+118.2x2x2+x+1 c)5x2-12(x2-6x+13)2dx 5x2-12(x2-6x+13)2=A2x-6+Bx2-6x+13+C2x-6+D(x2-6x+13)2 5x2-12=A2x-6+Bx2-6x+13+(C2x-6+D) A=0, B=5, C=15, D=13 5x2-6x+13+152x-6+13(x2-6x+13)2dx 5(x-3)2+4dx-15x2-6x+13+13(x-32+4)2dx 5x2-12(x2-6x+13)2dx=52.tan-1(x-22)-15x2-6x+13+138(tan-1x-222+x-3x26x+13) 5x2-12(x2-6x+13)2dx=5316tan-1(x-22)+18.13x-159x2-6x+13+C
d)x3(x2+2x+2)2dx
x3(x2+2x+2)2=A2X+2+Bx2+2x+2+C2X+2+E(x2+2x+2)2 x3=(A2X+2+B)( x2+2x+2)+ C2X+2+E RESOLVIENDO A=12, B=-3, C=1, E=2 (122X+2-3(x+1)2+1+2X+2+2((x+1)2+1)2)dx=12lnx2+2x+2-3tan1x+1+tan-1x+1+xx2+2x+2 x3(x2+2x+2)2dx=12lnx2+2x+2-2tan-1x+1+xx2+2x+2 e)dx(1+x2)4 x=tanθ;dx=secθ2dθ reemplazando secθ2dθsecθ8=cos6θdθ resolviendo dicha integral se obtiene que cos6θdθ=516θ+12senθcosθ+316senθcosθ(2cosθ2-1)-16(cosθsenθ)3 dx(1+x2)4=516tan-1x+12.xx2+1+316.xx2+1211+x2-1-16(xx2+1)3 dx(1+x2)4=516tan-1x+15x5+40x3+33x48x6+144x4+144x2+48+c
f)x3+x-1(x2+2)2dx x3+x-1(x2+2)2=A2X+Bx2+2+C2X+D(x2+2)2 x3+x-1=x2+2(A2X+B)+(C2X+D) RESOLVIENDO A=12, B=0 C=-12, D=-1 122Xx2+2+-122X-1(x2+2)2dx=12lnx2+2+12.1x2+2-1(x2+2)2dx x3+x-1(x2+2)2dx=12lnx2+2+12.1x2+2-(x4x2+2+28tan-1(2x2) x3+x-1(x2+2)2dx=12lnx2+2+2-x4x2+2-28tan-1(2x2)+c g)3x+5(2x2+x+1)2dx h)x9(x4-1)2dx dividiendo y por el algoritmo de la división se obtiene x9(x4-1)2=x+2x5-x(x4-1)2 2x5-x(x4-1)2=Ax-1+B(X1)2+C(X+1)+D(X+1)2+E2X+FX2+1+G2X+H(X2+1)2 resolviendo se obtiene que A=18, B=316, C=38, D=-116, E=-38, F=0,G=18, H=0 (x+38x-1+116(X-1)2+38X+1+-116X+12+-382XX2+1+182XX2+12)dx x9(x4-1)2dx=x22+38lnx-1-116.1x-1+38lnx+1+116.1x+1-38.lnx2+118.1x2+1+c
j)1(x2+9)3dx
