Question

$\int \frac{2 x+3}{(x-4)(5 x+2)} d x$

Name: Problem 27, Chapter 27: Engineering Mathematics
Uploaded: 2021-12-31T22:02:26-08:00
Duration: 1 min 29 s
Channel: Lucas Finney
Description: ∫(2 x+3)/((x-4)(5 x+2)) d x

   $\int \frac{2 x+3}{(x-4)(5 x+2)} d x$

Engineering Mathematics

K. A. Stroud, Dexter… 5th Edition

Chapter 27, Problem 27

Instant Answer

Solved by Expert Lucas Finney

12/31/2021

Step 1

We can write the integrand as: \[\frac{2x+3}{(x-4)(5x+2)} = \frac{A}{x-4} + \frac{B}{5x+2}\] Solving for A and B, we find that A = 1/2 and B = -1/2. So, we can rewrite the integral as: \[\int \frac{2x+3}{(x-4)(5x+2)} dx = \frac{1}{2}\int \frac{1}{x-4} dx - Show more…

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Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is a strategy for breaking down complex rational functions into a sum of simpler fractions with factors in the denominator. This method is particularly useful when the denominator can be factored into distinct or repeated linear or quadratic factors, making the subsequent integration process simpler.

Integration of Logarithmic Forms

Once a rational function is decomposed into simpler fractions, many terms take the form of 1/(ax+b) which, when integrated, results in a logarithmic function. Understanding how to integrate these standard forms is essential for solving the integrals that arise from the decomposition.

Rational Functions

A rational function is a quotient of two polynomials. In integration, these functions often require special techniques because direct integration is not straightforward, especially when the degree of the numerator is less than the degree of the denominator.

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