# How do you use partial fractions to find the integral #int (2x^3-4x^2-15x+5)/(x^2-2x-8)dx#?

See answer below:

Continuation:

By signing up, you agree to our Terms of Service and Privacy Policy

To integrate the given rational function using partial fractions, first factor the denominator. In this case, (x^2 - 2x - 8) factors as ((x - 4)(x + 2)). Then, express the given rational function as the sum of two fractions with undetermined coefficients, one for each factor in the denominator.

The partial fraction decomposition would look like this:

[\frac{2x^3 - 4x^2 - 15x + 5}{x^2 - 2x - 8} = \frac{A}{x - 4} + \frac{B}{x + 2}]

Multiplying both sides by (x^2 - 2x - 8) to clear the denominators, we get:

[2x^3 - 4x^2 - 15x + 5 = A(x + 2) + B(x - 4)]

Next, we can either equate coefficients or choose appropriate values of (x) to solve for (A) and (B). Choosing (x = 4), we eliminate the term containing (B), and choosing (x = -2), we eliminate the term containing (A).

[2(4)^3 - 4(4)^2 - 15(4) + 5 = A(4 + 2) \implies A = \frac{-43}{12}] [2(-2)^3 - 4(-2)^2 - 15(-2) + 5 = B(-2 - 4) \implies B = \frac{13}{12}]

Now, we integrate each term separately:

[\int \frac{-43}{12(x - 4)} + \frac{13}{12(x + 2)} dx]

This integrates to:

[\frac{-43}{12}\ln|x - 4| + \frac{13}{12}\ln|x + 2| + C]

Where (C) is the constant of integration.

By signing up, you agree to our Terms of Service and Privacy Policy

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.

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7