∫x4(x−1)(x2+1)dxx4(x−1)(x2+1)=Ax−1+Bx+Cx2+1⇒x4=A(x2+1)(Bx+C)(x−1)puttingx=1onbothsides1=2AA=12puttingx=0onbothsides0=A−cc=A=12puttingx=−1onbothsides1=2A+(−B+c)(−2)⇒1=1+2B−2c⇒2c=2BB=c=12so,∫x4(x−1)(x2+1)=12∫dxx−1+12∫x+1x2+1dx=12∫dxx−1+14∫2xx2+1dx+12∫dxx2+1=12log|x−1|+14log(x2+1)+12tan−1x+c