# How do you integrate #(2x^2+4x+12)/(x^2+7x+10)# using partial fractions?

The answer is

The numerator is

We perform a long division

Therefore

We factorise the denominator

We can perform the decomposition into partial fractions

The denominators are the same, we compare the numerators

So,

Therefore,

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To integrate the rational function ( \frac{2x^2 + 4x + 12}{x^2 + 7x + 10} ) using partial fractions, follow these steps:

- Factor the denominator (x^2 + 7x + 10): (x^2 + 7x + 10 = (x + 5)(x + 2)).
- Decompose the rational function into partial fractions: [ \frac{2x^2 + 4x + 12}{x^2 + 7x + 10} = \frac{A}{x + 5} + \frac{B}{x + 2} ]
- Multiply both sides by the denominator (x^2 + 7x + 10) to clear the fractions: [ 2x^2 + 4x + 12 = A(x + 2) + B(x + 5) ]
- Expand and collect like terms: [ 2x^2 + 4x + 12 = (A + B)x + 2A + 5B ]
- Equate coefficients of corresponding terms: [ A + B = 2 ] [ 2A + 5B = 12 ]
- Solve the system of equations to find the values of (A) and (B).
- Once you have (A) and (B), integrate each term separately: [ \int \frac{2x^2 + 4x + 12}{x^2 + 7x + 10} dx = \int \frac{A}{x + 5} dx + \int \frac{B}{x + 2} dx ] [ = A\ln|x + 5| + B\ln|x + 2| + C ]
- Substitute the values of (A) and (B) into the integral.
- Finally, add the constant of integration (C) if necessary.

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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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