How do you integrate #(x-1)/ (x^3 +x^2)# using partial fractions?
However, after simplifying,
When we compare the resp. co-effs. of the two sides, we obtain,
This well-known Rule has been applied in the last step:
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To integrate the given expression using partial fractions, first, factor the denominator:
[ x^3 + x^2 = x^2(x + 1) ]
Now, we express the integrand as a sum of partial fractions:
[ \frac{x - 1}{x^3 + x^2} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x + 1} ]
Multiplying both sides by the denominator ( x^3 + x^2 ) to clear the fractions:
[ x - 1 = A(x + 1) + Bx(x + 1) + Cx^2 ]
Now, equating coefficients:
For ( x ): [ 1 = A + B ]
For ( x^2 ): [ 0 = C + B ]
For the constant term: [ -1 = A ]
Solving these equations, we find: [ A = -1 ] [ B = 1 ] [ C = -1 ]
So, the partial fraction decomposition is: [ \frac{x - 1}{x^3 + x^2} = \frac{-1}{x} + \frac{1}{x^2} - \frac{1}{x + 1} ]
Now, we can integrate each term separately. The integrals are: [ \int \frac{-1}{x} dx = -\ln|x| ] [ \int \frac{1}{x^2} dx = -\frac{1}{x} ] [ \int \frac{-1}{x + 1} dx = -\ln|x + 1| ]
Therefore, integrating the original expression using partial fractions yields: [ -\ln|x| - \frac{1}{x} - \ln|x + 1| + C ] where ( C ) is the constant of integration.
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To integrate (x - 1) / (x^3 + x^2) using partial fractions, follow these steps:
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Factor the denominator if possible. In this case, the denominator can be factored as x^2(x + 1).
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Decompose the rational function into partial fractions. Assume the decomposition takes the form:
(x - 1) / (x^3 + x^2) = A / x + B / x^2 + C / (x + 1)
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Clear the fractions by multiplying both sides by the denominator:
(x - 1) = A(x^2)(x + 1) + B(x + 1) + C(x)(x^2)
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Expand and equate coefficients of like terms.
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Once you find the values of A, B, and C, rewrite the original integral using the partial fraction decomposition.
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Integrate each term separately.
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If needed, simplify the resulting expression.
By following these steps, you can integrate the given rational function using partial fractions.
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When evaluating a one-sided 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 one-sided 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 one-sided 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 one-sided 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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