# How do you integrate #(7x + 44)/(x^2 + 10x + 24)# using partial fractions?

x = -4 7(-4) + 44 = B(-4+6), B = 8

x = -6 7(-6) + 44 = A(-6+4), A = -1

so we have

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To integrate ( \frac{7x + 44}{x^2 + 10x + 24} ) using partial fractions, you first factor the denominator (x^2 + 10x + 24), which factors as ((x + 6)(x + 4)). Then, you decompose the rational function into partial fractions. Since the denominator factors into distinct linear factors, the partial fraction decomposition will be of the form ( \frac{A}{x + 6} + \frac{B}{x + 4} ).

Now, you solve for (A) and (B) by equating coefficients:

(7x + 44 = A(x + 4) + B(x + 6))

Expanding and collecting like terms:

(7x + 44 = (A + B)x + (4A + 6B))

By comparing coefficients of like terms:

(A + B = 7) (coefficients of (x))

(4A + 6B = 44) (constant terms)

Solving this system of linear equations gives (A = 2) and (B = 5).

Thus, the partial fraction decomposition of ( \frac{7x + 44}{x^2 + 10x + 24} ) is ( \frac{2}{x + 4} + \frac{5}{x + 6} ).

You can now integrate each term separately:

[ \int \frac{7x + 44}{x^2 + 10x + 24} , dx = \int \left(\frac{2}{x + 4} + \frac{5}{x + 6}\right) , dx ]

[ = 2\ln|x + 4| + 5\ln|x + 6| + C ]

where (C) is the constant of integration.

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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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