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How do you use partial fraction decomposition to decompose the fraction to integrate #(2x)/ (4x^2 + 12x + 9)#?

Answer 1

Ezra Adams

#(2x)/ (4x^2 + 12x + 9) = 1/(2x+3) - 3/(2x+3)^2#

#(2x)/ (4x^2 + 12x + 9) = (2x)/(2x+3)^2#

# = (2x+3-3)/(2x+3)^2#

# = (2x+3)/(2x+3)^2 -3/(2x+3)^2#

# = 1/(2x+3) - 3/(2x+3)^2#

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Answer 2

William Abdalla

To decompose the fraction (2x)/(4x^2 + 12x + 9) using partial fraction decomposition, first factor the denominator as (2x + 3)^2. Then, express the fraction as:

(2x)/(4x^2 + 12x + 9) = A/(2x + 3) + B/(2x + 3)^2

Next, find the values of A and B by equating the numerators:

2x = A(2x + 3) + B

Solve for A and B by comparing coefficients of like terms. After finding the values of A and B, rewrite the fraction with the decomposed form and 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.