How do you integrate # x^2/ [x^2+x+4]# using partial fractions?
where
First, put the partial fractions in writing.
Putting the partial fraction back in writing
Putting the partial fraction back in writing
Currently, executing the integration
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To integrate ( \frac{x^2}{x^2+x+4} ) using partial fractions, follow these steps:

First, factor the denominator ( x^2 + x + 4 ) if possible. In this case, it cannot be factored further over the real numbers.

Express ( \frac{x^2}{x^2+x+4} ) as a sum of partial fractions: ( \frac{x^2}{x^2+x+4} = \frac{A}{x  \alpha} + \frac{Bx + C}{x^2 + bx + c} ), where ( \alpha ) is a root of the denominator.

Multiply both sides of the equation by the denominator ( x^2 + x + 4 ) to clear the fractions.

Expand the resulting equation and collect like terms.

Equate coefficients of like terms on both sides of the equation.

Solve the resulting system of equations to find the values of ( A ), ( B ), and ( C ).

Once you have the values of ( A ), ( B ), and ( C ), rewrite the original integral using the partial fraction decomposition.

Now, integrate each term separately.

Finally, simplify the result if necessary.
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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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