Solve part c step by step. Solve part c step by step. (c) 1 + COSX Integrate : Integrate: x +3X+2 cos X COS X —cosx dx Evaluate the integral 3cos2x - l)cosx loge x (a) The value of the integral dx is: Share with your friends Share 0 Neha Sethi answered this ∫x3+3x+2x2+12x+1dxPerforming partial fractionx3+3x+2x2+12x+1=Bx+Ax2+1+Dx+Cx2+12+Ex+1x3+3x+2=Bx+Ax2+1x+1+Dx+Cx+1+Ex2+12x3+3x+2=Bx+Ax3+x2+x+1+Dx2+Dx+Cx+C+Ex4+E+2Ex2x3+3x+2=Bx4+Bx3+Bx2+Bx+Ax3+Ax2+Ax+A+Dx2+Dx+Cx+C+Ex4+E+2Ex2x3+3x+2=B+Ex4+B+Ax3+B+A+D+2Ex2+B+A+D+Cx+A+C+EOn comparing coeff. we getB+E=0B+A=1B+A+D+2E=0A+B+C+D=3A+C+E=2On solving we getA=12,B=12, C=2, D=0,E=-12So, ∫x3+3x+2x2+12x+1dx=12∫x+1x2+1dx+2∫1x2+12dx-12∫1x+1dxNow consider ∫x+1x2+1dx=∫xx2+1dx+∫1x2+1dx=lnx2+12+tan-1x+c1Now consider ∫1x2+12dxApply reduction formula∫1ax2+bndx=2n-32bn-1∫1ax2+bn-1dx+x2bn-1ax2+bn-1With a=1,b=1 and n=2=x2x2+1+12∫1x2+1dx=x2x2+1+tan-1x2+c2So, ∫x3+3x+2x2+12x+1dx=12∫x+1x2+1dx+2∫1x2+12dx-12∫1x+1dx=lnx2+14+3tan-1x2+xx2+1-lnx+12+C where C=c1+c2 0 View Full Answer