Evaluate the following integral:

Denominator is factorized, so let separate the fraction through partial fraction, hence let







By equating similar terms, we get


A + C = 0 A = – C ………..(iii)


B + D = 0 B = – D…………(iv)


– Ab2 – Ca2 = 1


– ( – C)b2 – Ca2 = 1 (from equation(iii))


…………..(v)


– b2B – a2D = 0


– b2( – D) – a2D = 0


D = 0


So equation(iv) becomes B = 0


So equation (iii) becomes,


We put the values of A, B, C, and D values back into our partial fractions in equation (i) and replace this as the integrand. We get





Split up the integral,



Let substitute


u = x2 – a2 du = 2dx


v = x2 – b2 dv = 2dx, so the above equation becomes,




On integrating we get



Substituting back, we get



Applying the logarithm rule we get



Note: the absolute value signs account for the domain of the natural log function (x>0).


Hence,



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