integrate x2 / {(x2+a2)(x2+b2)} w r t x - Maths - Integrals

Question

integrate x2 /
{(x2+a2)(x2+b2)} w r t
x

Vipin Verma · Accepted Answer

∫x2(x2+a2)(x2+b2)So add and substract a2 in the numeratorSo
∫x2+a2 -a2(x2+a2)(x2+b2) = ∫x2+a2
(x2+a2)(x2+b2)dx-∫a2(x2+a2)(x2+b2)dx = I1 -I2I1 =
∫dx(x2+b2) = 1btan-1xb (as dx(x2+a2) = 1atan-1xa)I2 =
∫a2dx(x2+a2)(x2+b2) =a2(b2-a2)∫{dx(x2+a2) -dx(x2+b2)} =
a2(b2-a2){1atan-1xa-1btan-1xb}So ∫x2dx(x2+a2)(x2+b2)
=1atan-1xa -[a2(b2-a2){1atan-1xa-1btan-1xb}]+C

integrate x2 / {(x2+a2)(x2+b2)} w.r.t x

integrate x² / {(x²+a²)(x²+b²)} w.r.t x