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How do you use partial fraction decomposition to decompose the fraction to integrate #(x^2-4)^-1#?

Answer 1

Cameron Alexander

#(x^2-4)^-1 is 1/(x^2-4)#

Its partial fraction would be #1/4[1/(x-2) -1/(x+2)]#

Integration would give #1/4[ln(x-2) -ln(x+2)}#

=#1/4 ln((x-2)/(x+2))#

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Answer 2

Isabella Ackerman

To decompose the fraction ( \frac{1}{x^2 - 4} ) using partial fraction decomposition, you would first factor the denominator ( x^2 - 4 ) as ( (x - 2)(x + 2) ). Then, you express the fraction as the sum of two simpler fractions with undetermined coefficients:

[ \frac{1}{x^2 - 4} = \frac{A}{x - 2} + \frac{B}{x + 2} ]

Where ( A ) and ( B ) are constants to be determined. Then, you solve for ( A ) and ( B ) by finding a common denominator and equating coefficients. Once you have ( A ) and ( B ), you integrate each term separately.

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Answer from HIX Tutor

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.