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 = 1⇒ B = 1 – D…………(iv)
25A + 4C = 0
⇒ 25( – C) + 4C = 0 (from equation(iii))
⇒ C = 0………..(v)
So equation(iv) becomes
So equation (iii) becomes, A = 0
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
in first partthe
in second parthe t
so the above equation becomes,
On integrating we get
(the standard integral of )
Substituting back, we get
Note: the absolute value signs account for the domain of the natural log function (x>0).
Hence,