and also (and i checked with a calculator) it is NOT true that [ 5x-7]/(x^2) + 12/(x+2) = [(5x^2)+(3x)-(2)] / [(x^3)+(2x^2)]. the way i did the decomposition was into the termsa/x+b/x^2+c/(x+2)which i solved the constants a, b, and c for to find that the integral was equal to∫(2/x-1/x^2+3/(x+2)) which i also check with a calculator and found to be right. next, integrate like you did:2ln|x|+1/x+3ln|x+2|+c-The method is not invalid.......
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5x^2 + 3x - 2 = (B + C)x^2 + (A + 2B)x + 2A
Equating like coefficients,
B + C = 5
A + 2B = 3
2A = -2
So,
A = -1
B = 2
C = 3
And, our partial fraction expansion of f(x) is:
f(x) = -1/x^2 + 2/x + 3/(x + 2)
Integrating term-by-term,
∫ f(x) dx
= ∫ -1/x^2 dx + ∫ 2/x dx + ∫ 3/(x + 2) dx
= -∫ 1/x^2 dx + 2 ∫ 1/x dx + 3 ∫ 1/(x + 2) dx
= 1/x + 2 ln x + 3 ln (x + 2) + C
i think you just solved for the constants wrong, and also (and i checked with a calculator) it is NOT true that [ 5x-7]/(x^2) + 12/(x+2) = [(5x^2)+(3x)-(2)] / [(x^3)+(2x^2)]. the way i did the decomposition was into the terms
a/x+b/x^2+c/(x+2)
which i solved the constants a, b, and c for to find that the integral was equal to
∫(2/x-1/x^2+3/(x+2)) which i also check with a calculator and found to be right. next, integrate like you did:
2ln|x|+1/x+3ln|x+2|+c
The method is not invalid. Your partial fractions are incorrect, try x=4 I get two different values.
