How do you use partial fraction decomposition to decompose the fraction to integrate #(x^25x+6)/(x^3x^2+2x)#?
See below
The partial fraction decomposition is:
If you have time check by combining the fractions on the left to make sure you've made no mistakes.
Now integrate:
The first integral is easy. I'd split the second using:
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To decompose the fraction ((x^2  5x + 6) / (x^3  x^2 + 2x)) using partial fraction decomposition, follow these steps:

Factor the denominator (x^3  x^2 + 2x) if possible. In this case, the denominator doesn't factor further.

Write the fraction as a sum of simpler fractions with unknown numerators:
(\frac{x^2  5x + 6}{x^3  x^2 + 2x} = \frac{A}{x} + \frac{B}{x  2} + \frac{C}{x + 1})
 Clear the denominators by multiplying both sides of the equation by the original denominator (x^3  x^2 + 2x):
(x^2  5x + 6 = A(x  2)(x + 1) + Bx(x + 1) + Cx(x  2))
 Expand the right side and collect like terms:
(x^2  5x + 6 = A(x^2  x  2) + B(x^2 + x) + C(x^2  2x))
 Equate the coefficients of like terms on both sides of the equation:
For (x^2) terms: (1 = A + B + C) For (x) terms: (5 = A + B  2C) For constant terms: (6 = 2A)
 Solve the system of equations for (A), (B), and (C).
Solving these equations will give you the values of (A), (B), and (C). Once you find those values, substitute them back into the partial fraction decomposition:
(\frac{x^2  5x + 6}{x^3  x^2 + 2x} = \frac{3}{x} + \frac{2}{x  2} + \frac{1}{x + 1})
After decomposing the fraction, you can integrate each term separately.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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