So I'm trying to integrate a function using partial fractions. Here is the integral of interest: ∫(3x^2+5x+3)/[(x+2)(x^2+1)]dx.
Since the numerator's degree of the polynomial is lesser than that of the denominator's degree, it is clear to separate. However, once I separate and multiply like terms, I encounter obstacle when determining the new scalar numerators.
(1) ∫(3x^2+5x+3)/[(x+2)(x^2+1)]dx = ∫[A/(x+2)]+[Bx+C/(x^2+1)]dx.
(2) Multiply for common denominator to obtain numerator corresponding to (A+B)x^2+(2B+C)x+(A+2C). Solving A, B, and C for common roots from the initial polynomial, I set up (A+B) = 3, (2B+C) = 5, (A+2C)=3. My new terms are now: A=1; B=2; C=1.
(3) ∫dx/(x+2) +∫(2x+1)/(x^2+1)dx is now the new, separated function. ∫dx/(x+2) will integrate into ln|x+2|+C. Further separation is needed for the second integral, ∫(2x+1)/(x^2+1)dx = ∫2x/(x^2+1)dx +∫dx/(x^2+1). ∫dx/(x^2+1) integrates nicely into arctanx. ∫2x/(x^2+1)dx needs substitution however, which eliminates the 2x in the numerator without adding any scalars: thus ∫2x/(x^2+1)dx = ln|x^2+1|.
∫(3x^2+5x+3)/[(x+2)(x^2+1)]dx = ln|(x+2)(x^2+1)|+arctanx+C !
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