Evaluate : ∫(x^2+1)/(x^2+4)(x^2+25)dx

Question

Evaluate : \(\int\frac{x^2+1}{(x^2+4)(x^2+25)}dx\)

Answer 1

Let I = \(\int\frac{x^2+1}{(x^2+4)(x^2+25)}dx\)

Put x² = y

⇒ \(\frac{x^2+1}{(x^2+4)(x^2+25)}\) = \(\frac{y+1}{(y+4)(y+25)}\)

Now,

\(\frac{y+1}{(y+4)(y+25)}\) = \(\frac{A}{y+4}+\)\(\frac{B}{y+25}\)

⇒ \(\frac{y+1}{(y+4)(y+5)}\) = \(\frac{A(y+25)+B(y+4)}{(y+4)(y+5)}\)

⇒ y + 1  = (A + B)y + (25 A + AB)

Equating coefficients, we get

A + B = 1 and 25 A + 4B = 1

⇒ A = \(\frac{-1}{7},\) B = \(\frac{8}{7}\)