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)


3A + C = 2


3( – C) + C = 2 (from equation(iii))


C = – 1


So equation(iii) becomes A = 1


And also 3B + D = 0 (from equation (ii))


3( – D) + D = 0 (from equation (iv))


D = 0


So equation (iv) becomes, B = 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




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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