How do you integrate #(2x) / (4x^2 + 12x + 9)# using partial fractions?

Answer 1

#=1/2ln|2x+3|+3/(2(2x+3))+C#.
You don't use partial fractions, because the denominator is a perfect square.

#int (2x)/(4x^2+12x+9)dx# #=int(2x)/(2x+3)^2dx# So substitute #u=2x+3#, #dx=(du)/2#: #=1/2int (u-3)/u^2du# #=1/2intu^-1-3u^-2dx# #=1/2ln|2x+3|+3/(2(2x+3))+C#
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Answer 2

To integrate ( \frac{2x}{4x^2 + 12x + 9} ) using partial fractions, follow these steps:

  1. Factor the denominator ( 4x^2 + 12x + 9 ) into its irreducible quadratic factors.
  2. Write the fraction ( \frac{2x}{4x^2 + 12x + 9} ) as a sum of partial fractions, where the numerator of each fraction is a constant or a linear polynomial, and the denominator is one of the irreducible quadratic factors.
  3. Solve for the unknown coefficients by equating the original fraction to the sum of the partial fractions.
  4. Once you have found the partial fraction decomposition, integrate each term separately.
  5. Combine the integrals of the partial fractions to obtain the final result.

For detailed step-by-step instructions on how to perform partial fraction decomposition and integrate each term, you can refer to calculus textbooks or online resources covering the topic of partial fractions.

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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.

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.

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.

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.

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