Question

Use partial fraction decomposition to integrate the following function. f(x)= fx²+mx+l/(lx-2)²(x²-fx-2)

Answer 1

The partial fraction decomposition of the given function
`f(x) = (x^2 + mx + l) / ((lx - 2)^2 * (x^2 - fx - 2))` is:

`\[ (A)/((lx - 2)) + (B)/((lx - 2)^2) + (Cx + D)/((x^2 - fx - 2)) + (Ex + F)/((x^2 - fx - 2)^2) \]`

Where A, B, C, D, E, and F are constants.

The integral of f(x) can be expressed as the sum of integrals of each partial fraction. After finding the values of the constants, the integration can be carried out, resulting in the final answer.

Step-by-step explanation:

To decompose the given rational function into partial fractions, we express it as a sum of simpler fractions. The denominator factors are
`(lx - 2)^2 and (x^2 - fx - 2)`. Therefore, we express the partial fractions as:

`\[ (A)/((lx - 2)) + (B)/((lx - 2)^2) + (Cx + D)/((x^2 - fx - 2)) + (Ex + F)/((x^2 - fx - 2)^2) \]`

Next, we find the values of A, B, C, D, E, and F by equating the numerators and simplifying. Once the constants are determined, the integration can be performed by taking the antiderivative of each term. The integral of each partial fraction can be found separately, leading to the final solution.

This process involves algebraic manipulations and integration techniques, ensuring a systematic approach to finding the integral of the given function. The resulting expression will be the antiderivative of the original function, expressed in terms of the constants obtained during the partial fraction decomposition.