How do you use partial fraction decomposition to decompose the fraction to integrate #1/((x+2)(x^2+4))#?
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To decompose the fraction ( \frac{1}{(x+2)(x^2+4)} ) using partial fraction decomposition:

Factor the denominator completely: ( (x+2)(x^2+4) ).

Write the fraction as a sum of simpler fractions with unknown numerators: [ \frac{1}{(x+2)(x^2+4)} = \frac{A}{x+2} + \frac{Bx+C}{x^2+4} ]

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

Expand and simplify the equation obtained in step 3.

Equate coefficients of like terms on both sides of the equation to find the values of ( A ), ( B ), and ( C ).

Once you find the values of ( A ), ( B ), and ( C ), substitute them back into the decomposition:
[ \frac{1}{(x+2)(x^2+4)} = \frac{A}{x+2} + \frac{Bx+C}{x^2+4} ]
 Now, integrate each term separately.
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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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