Q.462. Integration.

Question 462.
Find  (x^3) / [(x^2) + 4x + 8] dx.

Answer 462.
Note that the numerator is a polynomial of degree 3 and the denominator is of degree 2.
Hence, the first step is to perform a division and express the function in the form of a quotient + remainder/denominator so that the remainder is a polynomial of degree less than the denominator. Integration follows thereafter. Thus,
x^3
= x (x^2 + 4x + 8) - 4x^2 - 8x
= x (x^2 + 4x + 8) - 4(x^2 + 4x + 8) + 8x + 32

=> x^3/(x^2 + 4x + 8)
= (x - 4) + 8 (x + 4) / (x^2 + 4x + 8)

=> Integral
= ∫ (x - 4) dx + 4 ∫ (2x + 4 + 4) / (x^2 + 8x + 8) dx
= x^2/2 - 4x + 4 ∫ (d(x^2 + 8x + 8) / (x^2 + 8x + 8) + 16 ∫ dx / [(x + 2)^2 + 2^2]
= x^2/2 - 4x + 4 log(x^2 + 8x + 8) + 8 tan^-1 [(x + 2)/2] + c.

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