Evaluate the following integral:
The denominator is factorized, so let separate the fraction through partial fraction, hence let
⇒ 2x – 3 = A(x + 1)(2x + 3) + B(x – 1)(2x + 3) + C(x – 1)(x + 1)……(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 = – 1 in the above equation, we get
⇒ 2( – 1) – 3 = A(( – 1) + 1)(2( – 1) + 3) + B(( – 1) – 1)(2( – 1) + 3) + C(( – 1) – 1)(( – 1) + 1)
⇒ – 5 = 0 – 2B + 0
Now put x = 1 in equation (ii), we get
⇒ 2(1) – 3 = A((1) + 1)(2(1) + 3) + B((1) – 1)(2(1) + 3) + C((1) – 1)((1) + 1)
⇒ – 1 = 10A + 0 + 0
Now put in equation (ii), we get
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 + 1 ⇒ du = dx,
y = x – 1 ⇒ dy = dx and
so the above equation becomes,
On integrating we get
Substituting back, we get
Note: the absolute value signs account for the domain of the natural log function (x>0).
Hence,