A step-by-step answer would be awesome! I'm studying for an exam >_< Thanks!
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Since the denominator factors to x^2(x - 3), split the integrand up into Partial Fractions.
(5x^2 - 10x + 3)/(x^2 - 3x^2) = A/x + B/x^2 + C/(x - 3).
Multiplying both sides by x^2(x - 3) gives:
5x^2 - 10x + 3 = Ax(x - 3) + B(x - 3) + Cx^2.
By expanding the right side:
Ax(x - 3) + B(x - 3) + Cx^2 = (Ax^2 - 3Ax) + (Bx - 3B) + Cx^2
= (A + C)x^2 + (-3A + B)x - 3B.
Then, by comparing coefficients:
(i) By comparing constants: -3B = 3 ==> B = -1
(ii) By comparing x coefficients: -3A + B = -10 ==> -3A - 1 = -10 ==> A = 3
(iii) By comparing x^2 coefficients: A + C = 5 ==> C + 3 = 5 ==> C = 2.
Thus:
(5x^2 - 10x + 3)/(x^2 - 3x^2) = 3/x - 1/x^2 + 2/(x - 3).
Integrating term-by-term therefore yields:
∫ (5x^2 - 10x + 3)/(x^3 - 3x^2) dx
= ∫ [3/x - 1/x^2 + 2/(x - 3)] dx
= 3 ∫ 1/x dx - ∫ 1/x^2 dx + 2 ∫ 1/(x - 3) dx
= 3ln|x| + 1/x + 2ln|x - 3| + C.
I hope this helps!
(5x^2 - 10x + 3)/(x^2 - 3x^2) = A/x + B/x^2 + C/(x - 3).
Multiplying both sides by x^2(x - 3) gives:
5x^2 - 10x + 3 = Ax(x - 3) + B(x - 3) + Cx^2.
By expanding the right side:
Ax(x - 3) + B(x - 3) + Cx^2 = (Ax^2 - 3Ax) + (Bx - 3B) + Cx^2
= (A + C)x^2 + (-3A + B)x - 3B.
Then, by comparing coefficients:
(i) By comparing constants: -3B = 3 ==> B = -1
(ii) By comparing x coefficients: -3A + B = -10 ==> -3A - 1 = -10 ==> A = 3
(iii) By comparing x^2 coefficients: A + C = 5 ==> C + 3 = 5 ==> C = 2.
Thus:
(5x^2 - 10x + 3)/(x^2 - 3x^2) = 3/x - 1/x^2 + 2/(x - 3).
Integrating term-by-term therefore yields:
∫ (5x^2 - 10x + 3)/(x^3 - 3x^2) dx
= ∫ [3/x - 1/x^2 + 2/(x - 3)] dx
= 3 ∫ 1/x dx - ∫ 1/x^2 dx + 2 ∫ 1/(x - 3) dx
= 3ln|x| + 1/x + 2ln|x - 3| + C.
I hope this helps!
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Did I now, sage?
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It's a partial fractions problem. The numerator has a degree less than the denominator (degree 2 vs. degree 3) so the fraction is proper; we can jump right into things by factoring the denominator and setting up the form of the answer by following the rules of partial fractions (which you can find in your book, I'm sure).
x^3 -3x^2 factors to x^2(x-3) so we'd assume the answer is of the form
A/x + B/x^2 + C/(x-3) (or (Ax+B) / x^2 + C/(x-3) if you'd prefer)
This would mean
Ax(x-3) + B(x-3) + Cx^2 would have to equal 5x^2 -10x +3
There are several ways to go about finding A, B and C. I'll use convenient values for a while:
if x = 3 then we have
A3(3-3) + B(3-3) + C3^2 = 5(3)^2-10(3) + 3
9C = 45-30+3 = 18
C = 2
if x = 0 then we have
A0(0-3) + B(0-3) + C0^2 = 5(0)^2-10(0) + 3
-3B = 3
B = -1
By a bit a trickiness, I can see that Ax^2 + Cx^2 has to equal 5x^2. Since C =2, A = 3
Therefore, we want to integrate
3 / x -1/x^2 + 2/(x-3)
3 ln |x| + 1/x + 2 ln |x-3| + C (the arbitrary C of integrals, not the C from above).
x^3 -3x^2 factors to x^2(x-3) so we'd assume the answer is of the form
A/x + B/x^2 + C/(x-3) (or (Ax+B) / x^2 + C/(x-3) if you'd prefer)
This would mean
Ax(x-3) + B(x-3) + Cx^2 would have to equal 5x^2 -10x +3
There are several ways to go about finding A, B and C. I'll use convenient values for a while:
if x = 3 then we have
A3(3-3) + B(3-3) + C3^2 = 5(3)^2-10(3) + 3
9C = 45-30+3 = 18
C = 2
if x = 0 then we have
A0(0-3) + B(0-3) + C0^2 = 5(0)^2-10(0) + 3
-3B = 3
B = -1
By a bit a trickiness, I can see that Ax^2 + Cx^2 has to equal 5x^2. Since C =2, A = 3
Therefore, we want to integrate
3 / x -1/x^2 + 2/(x-3)
3 ln |x| + 1/x + 2 ln |x-3| + C (the arbitrary C of integrals, not the C from above).
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.. (5x² - 10x + 3) / (x^3 - 3x²)
= -1/x² + 3/x + 2/(x - 3)
.. ∫ [ (5x² - 10x + 3) / (x^3 - 3x²) ] dx
= - ∫ (1/x²) dx + 3∫ (1/x) dx + 2 ∫ (1/(x - 3) ) dx
= 1/x + 3 ln(x) + 2 ln(x - 3) + constant
*** Don't bother to choose a best, Brian will do that for you.
= -1/x² + 3/x + 2/(x - 3)
.. ∫ [ (5x² - 10x + 3) / (x^3 - 3x²) ] dx
= - ∫ (1/x²) dx + 3∫ (1/x) dx + 2 ∫ (1/(x - 3) ) dx
= 1/x + 3 ln(x) + 2 ln(x - 3) + constant
*** Don't bother to choose a best, Brian will do that for you.