Question

integrate (x2+1)/(x2+4)(x2+25)

Answer 1

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SOLUTION:--

Let I = ∫(x2+1)(x2+4)(x2+25) dxput x2 = y, then(x2+1)(x2+4)(x2+25) = y+1(y+4)(y+25)Let y+1(y+4)(y+25) = Ay+4 + By+25⇒y+1 = (A+B)y + (25A+4B)Comparing the coefficient of y on both sides, we getA + B = 1       .......(1)Comparing constants on both sides, we get25A + 4B = 1     .......(2)Solving (1) and (2), we getA = −17   and   B = 87Now, y+1(y+4)(y+25) = −17×1y+4 + 87×1y+25⇒(x2+1)(x2+4)(x2+25) = −17×1x2+4 + 87×1x2+25⇒∫(x2+1) dx(x2+4)(x2+25) = −17∫dxx2+4 + 87∫dxx2+25⇒∫(x2+1) dx(x2+4)(x2+25)= −17∫dxx2+(2)2 + 87∫dxx2+(5)2⇒∫(x2+1) dx(x2+4)(x2+25) = −17 ×12tan−1(x2) + 87×15tan−1(x5) + C⇒∫(x2+1) dx(x2+4)(x2+25)= −114tan−1(x2) + 835tan−1(x5) + C