Find Integral(2x)By((Xsquare2 + 1)(Xpower4 + 4))Dx - Mathematics

Find int (2x)/((x^2 + 1)(x^4 + 4))dx

Solution

Let x2  = t

2xdx = dt

=> dx = dt/(2x)

I = int (2x)/((x^2 + 1)(x^4 + 4)) dx

= int (dt)/((t +1)(t^2 + 4))

Let 1/((t + 1)(t^2 + 4)) = A/(t + 1) + (Bt + C)/(t^2+4) .....(1)

=> 1 = A(t^2 + 4) + (Bt + C) (t + 1) ...(2)

Putting t=−1 in (2)

1 = A(1 + 4) +0

⇒5A = 1

=> A = 1/5

Putting t = 0 in 2

4A + C = 1

=>C = 1 - 4/5

=> C = 1/5

Putting t = 1 in (2)

1 = 5A + 2B + 2C

=> -2B = 2/5

=>B = (-1)/5

Putting the values of A, B and C in (1)

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