# How do you integrate #(5x)/(2x^2+11x+12)# using partial fractions?

We start by factoring the denominator.

The partial fraction decomposition will therefore be of the form:

We now write a systems of equations.

Solve:

Hopefully this helps!

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To integrate the rational function (5x)/(2x^2 + 11x + 12) using partial fractions, follow these steps:

- Factor the denominator: 2x^2 + 11x + 12 = (2x + 3)(x + 4).
- Decompose the rational function into partial fractions: (5x)/(2x^2 + 11x + 12) = A/(2x + 3) + B/(x + 4).
- Multiply both sides by the denominator to clear the fractions: 5x = A(x + 4) + B(2x + 3).
- Expand and collect like terms: 5x = (A + 2B)x + (4A + 3B).
- Equate the coefficients of like terms on both sides to find the values of A and B.
- Solve the system of equations to find the values of A and B.
- Once you have A and B, integrate each term separately.
- The integral of (5x)/(2x^2 + 11x + 12) is A ln|2x + 3| + B ln|x + 4| + C, where C is the constant of integration.

After solving for A and B, the integral would be: A ln|2x + 3| + B ln|x + 4| + C.

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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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