How do you use partial fraction decomposition to decompose the fraction to integrate #(7)/(x^2+13x+40)#?
The integral equals
Now we can decompose in partial fractions.
Now we have a system of equations.
Substituting the first equation into the second we see that
Hopefully this helps!
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To decompose the fraction (7)/(x^2 + 13x + 40) using partial fraction decomposition, follow these steps:

Factor the denominator: x^2 + 13x + 40 = (x + 8)(x + 5).

Write the fraction as a sum of two simpler fractions with undetermined coefficients: (7)/(x^2 + 13x + 40) = A/(x + 8) + B/(x + 5).

Multiply both sides of the equation by (x + 8)(x + 5) to clear the denominators: 7 = A(x + 5) + B(x + 8).

Expand and collect like terms: 7 = (A + B)x + (5A + 8B).

Equate coefficients of like terms on both sides of the equation: For x terms: A + B = 0. For constant terms: 5A + 8B = 7.

Solve the system of equations to find the values of A and B.

Once you find the values of A and B, substitute them back into the partial fraction decomposition.

Now integrate each term separately.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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