Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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How do we solve for B and D in this problem? I was able to get A and C but couldn't figure out B And D 

The image displays a mathematical solution involving partial fraction decomposition and integration.

The problem is labeled as "8.5.57" and involves expressing the fraction \(\frac{x^3 + 5x}{(x^2+3)^2}\) as a sum of simpler fractions:

\[
\frac{x^3 + 5x}{(x^2+3)^2} = \frac{Ax+B}{x^2+3} + \frac{Cx+D}{(x^2+3)^2}
\]

To solve for \(A\), \(B\), \(C\), and \(D\), we equate:

\[ 
x^3 + 5x = (Ax + B)(x^2 + 3) + Cx + D = Ax^3 + Bx^2 + (3A + C)x + 3B + D 
\]

By comparing coefficients, we find:

- \(A = 1\)
- \(B = 0\)
- \(3A + C = 5\)
- \(3B + D = 0\)

Thus, solving these gives:

- \(C = 2\)
- \(D = 0\)

The integral of the original function is then evaluated:

\[
\int \frac{x^3 + 5x}{(x^2+3)^2} \, dx = \int \left( \frac{x}{x^2+3} + \frac{2x}{(x^2+3)^2} \right) \, dx 
\]

This results in:

\[
= \frac{1}{2} \ln |x^2+3| - \frac{1}{x^2+3} + C
\] 

This step-by-step process explains how the integration is carried out.
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Transcribed Image Text:The image displays a mathematical solution involving partial fraction decomposition and integration. The problem is labeled as "8.5.57" and involves expressing the fraction \(\frac{x^3 + 5x}{(x^2+3)^2}\) as a sum of simpler fractions: \[ \frac{x^3 + 5x}{(x^2+3)^2} = \frac{Ax+B}{x^2+3} + \frac{Cx+D}{(x^2+3)^2} \] To solve for \(A\), \(B\), \(C\), and \(D\), we equate: \[ x^3 + 5x = (Ax + B)(x^2 + 3) + Cx + D = Ax^3 + Bx^2 + (3A + C)x + 3B + D \] By comparing coefficients, we find: - \(A = 1\) - \(B = 0\) - \(3A + C = 5\) - \(3B + D = 0\) Thus, solving these gives: - \(C = 2\) - \(D = 0\) The integral of the original function is then evaluated: \[ \int \frac{x^3 + 5x}{(x^2+3)^2} \, dx = \int \left( \frac{x}{x^2+3} + \frac{2x}{(x^2+3)^2} \right) \, dx \] This results in: \[ = \frac{1}{2} \ln |x^2+3| - \frac{1}{x^2+3} + C \] This step-by-step process explains how the integration is carried out.
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