How do you integrate #int (x+4) / [(x1)(x^2+4)]# using partial fractions?
The integrand's partial fraction decomposition will take the following form:
And so,
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To integrate (\int \frac{x + 4}{(x  1)(x^2 + 4)} ,dx) using partial fractions, follow these steps:

Perform partial fraction decomposition to express the integrand as a sum of simpler fractions.

Start by expressing (\frac{x + 4}{(x  1)(x^2 + 4)}) as: [\frac{x + 4}{(x  1)(x^2 + 4)} = \frac{A}{x  1} + \frac{Bx + C}{x^2 + 4}]

Clear the denominators by multiplying both sides of the equation by ((x  1)(x^2 + 4)): [x + 4 = A(x^2 + 4) + (Bx + C)(x  1)]

Expand and collect like terms on the right side of the equation.

Equate the coefficients of corresponding 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 determined the values of (A), (B), and (C), rewrite the original integral with the partial fraction decomposition.

Integrate each term separately.

Finally, sum up the integrals of each term to obtain the overall result.
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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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