# How do you integrate #(x^2-2)/((x+1)(x^2+3))# using partial fractions?

Please see the explanation.

Expand:

Let x = -1 to make B and C disappear:

Let x = 0 to make B disappear:

Let x = 1:

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To integrate (\frac{{x^2 - 2}}{{(x + 1)(x^2 + 3)}}) using partial fractions, first express the fraction as a sum of simpler fractions. The general form of partial fractions for a quadratic denominator is (\frac{{Ax + B}}{{(ax^2 + bx + c)}}). Therefore:

(\frac{{x^2 - 2}}{{(x + 1)(x^2 + 3)}} = \frac{{A}}{{x + 1}} + \frac{{Bx + C}}{{x^2 + 3}})

To solve for (A), (B), and (C), multiply both sides by the common denominator ((x + 1)(x^2 + 3)), then compare coefficients.

(x^2 - 2 = A(x^2 + 3) + (Bx + C)(x + 1))

Expand and equate coefficients:

(x^2 - 2 = Ax^2 + 3A + Bx^2 + Bx + Cx + C)

Comparing coefficients of like terms:

For (x^2): (1 = A + B)

For (x): (0 = B + C)

For the constant term: (-2 = 3A + C)

Solving these equations simultaneously yields the values of (A), (B), and (C). Then, integrate each term separately.

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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.

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