# What is #int (x^3-2x^2+6x+9 ) / (2x^2- x +3 )#?

Since the quadratic on the denominator is irreducible, we can write this quotient in the form of a partial fraction decomposition.

When this fraction is simplified, it yields

Through coeffecient equations,

Then

One significant general integral finding (particularly in the context of rational functions) is that,

We may state,

After that, switching,

The last integral to evaluate is the arctangent integral, which can only be solved by completing the square because the quadratic on the denominator is irreducible (no real factors).

It is a typical outcome that,

Applying the inverse chain rule to this outcome yields

At last,

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To find the integral of ( \frac{x^3 - 2x^2 + 6x + 9}{2x^2 - x + 3} ), you would typically use the method of partial fraction decomposition followed by integration. However, since the partial fraction decomposition can be quite complex for this expression, it would be necessary to perform the division first to simplify the expression before attempting integration.

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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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