Question

Solve the following integral  ∫ (x^2 + x + 2) / (x + 1)(x^2 + 1) dx. Use partial fraction decomposition. Show all work and steps to get to solution.

Answer 1

we have to evaluate,

using partial fraction we can write,

-----------------------------------------------------------------------1)

------------------------------------------------------2)

Put x = - 1 in equation 2) we can write,

Put A = 1 in equation 2) we can write,

  ------------------------------------------------3)

equate co efficient of x² on both the side of equation 3) we can write,

equate constant of both the side of equation 3) we can write,

Put A = 1, B = 0 and C = 1 in equation 1) we can write,

Hence we can say that,

-------------------------------------------------4)

we will evaluate,

Let's assume that x + 1 = u hence,

Hence,

substitute back u = x + 1 hence,

we will evaluate,

we can write,

we have,

Put them in equation 4) we can write,

add constant to the solution we can write,