Q. 205.0( 2 Votes )

Evaluate the foll

Answer :

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






x2 + 6x – 8 = A(x – 2)(x + 2) + Bx(x + 2) + Cx(x – 2)……(ii)


We need to solve for A, B and C. One way to do this is to pick values for x which will cancel each variable.


Put x = 0 in the above equation, we get


02 + 6(0) – 8 = A(0 – 2)(0 + 2) + B(0)(0 + 2) + C(0)(0 – 2)


– 8 = – 4A + 0 + 0


A = 2


Now put x = 2 in equation (ii), we get


22 + 6(2) – 8 = A(2 – 2)(2 + 2) + B(2)(2 + 2) + C(2)(2 – 2)


8 = 0 + 8B + 0


B = 1


Now put x = – 2 in equation (ii), we get


( – 2)2 + 6( – 2) – 8 = A(( – 2) – 2)(( – 2) + 2) + B( – 2)(( – 2) + 2) + C( – 2)(( – 2) – 2)


– 16 = 0 + 0 + 8C


C = – 2


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





Split up the integral,



Let substitute


u = x – 2 du = dx,


y = x + 2 dy = dx, so the above equation becomes,



On integrating we get



Substituting back, we get



Applying 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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