Question

(x⁴+2x²+3)/(x²+1)⁵ resolve this into partial fractions​

Answer 1

Step-by-step explanation:

x4−2x2+4x+1x3−x2−x+1

=x4−x3−x2+x+x3−x2+3x+1x3−x2−x+1

=x(x3−x2−x+1)+x3−x2+3x+1x3−x2−x+1

=x+x3−x2+3x+1x3−x2−x+1

=x+x3−x2−x+1+4xx3−x2−x+1

=x+1+4xx3−x2−x+1

=x+1+4xx2(x−1)−1(x−1)

=x+1+4x(x−1)(x2−1)

=x+1+4x(x+1)(x−1)2(1)

Let4x(x+1)(x−1)2=Ax+1+Bx−1+C(x−1)2

⟹4x=A(x−1)2+B(x2−1)+C(x+1)

When x = -1

−4=4A⟹A=−1

When x = 1

4=2C⟹C=2

Comparing coefficient ofx2 on both sides

A+B=0⟹B=−A=1

⟹4x(x+1)(x−1)2=−1x+1+1x−1+2(x−1)2

Putting this result in (1)

x4−2x2+4x+1x3−x2−x+1=x+1−1x+1+1x−1+2(x−1)2

∫x4−2x2+4x+1x3−x2−x+1dx

=∫(x+1−1x+1+1x−1+2(x−1)2)dx

=x22+x−ln(x+1)+ln(x−1)−2x−1+c