# How do you use partial fraction decomposition to decompose the fraction to integrate #x^2/(x^2+x+4)#?

See the explanation section, below.

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To decompose the fraction ( \frac{x^2}{x^2 + x + 4} ) using partial fraction decomposition, first, factor the denominator. If the denominator factors into distinct linear factors or repeated linear factors, you can proceed with partial fraction decomposition.

The denominator ( x^2 + x + 4 ) does not factor further into linear factors with real coefficients. Hence, we'll consider complex roots.

The roots can be found using the quadratic formula:

[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} ]

For the quadratic equation ( x^2 + x + 4 = 0 ), where ( a = 1 ), ( b = 1 ), and ( c = 4 ):

[ x = \frac{{-1 \pm \sqrt{{1^2 - 4 \cdot 1 \cdot 4}}}}{{2 \cdot 1}} ]

[ x = \frac{{-1 \pm \sqrt{{1 - 16}}}}{{2}} ]

[ x = \frac{{-1 \pm \sqrt{{-15}}}}{{2}} ]

Since the discriminant is negative, the roots will be complex:

[ x = \frac{{-1 \pm i\sqrt{{15}}}}{{2}} ]

Thus, the partial fraction decomposition will involve terms with complex coefficients. You would express ( \frac{x^2}{x^2 + x + 4} ) as a sum of fractions, where each denominator corresponds to a linear factor of the original denominator. In this case, since the denominator has no real linear factors, you would decompose it into terms involving the complex roots found.

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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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