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,