Sure! Let's break down the process to find the solution step-by-step.
Step 1: Setup the Partial Fraction Decomposition
We start by expressing the given rational function [tex]\(\frac{2y^3 + 17y^2 + 44y + 61}{(y+5)^2(y+1)^2}\)[/tex] in terms of its partial fraction decomposition:
[tex]\[
\frac{2y^3 + 17y^2 + 44y + 61}{(y+5)^2(y+1)^2} = \frac{A}{y+5} + \frac{B}{y+1} + \frac{C}{(y+5)^2} + \frac{D}{(y+1)^2}
\][/tex]
Step 2: Combine the Fractions
To combine these fractions, we need a common denominator [tex]\((y+5)^2(y+1)^2\)[/tex]:
[tex]\[
\frac{A}{y+5} + \frac{B}{y+1} + \frac{C}{(y+5)^2} + \frac{D}{(y+1)^2} = \frac{A(y+1)^2 + B(y+5)^2 + C(y+1)^2(y+5) + D(y+5)^2(y+1)}{(y+5)^2(y+1)^2}
\][/tex]
Step 3: Equate the Numerators
Now, equate the numerators of both sides:
[tex]\[
2y^3 + 17y^2 + 44y + 61 = A(y+1)^2 + B(y+5)^2 + C(y+1)^2(y+5) + D(y+5)^2(y+1)
\][/tex]
Step 4: Solve for Coefficients (A, B, C, D)
To find [tex]\(A\)[/tex], [tex]\(B\)[/tex], [tex]\(C\)[/tex], and [tex]\(D\)[/tex], we solve this system of equations for each term. After solving the equations, we obtain:
[tex]\[
A = 2.0, \quad B = 0.0, \quad C = 1.0, \quad D = 2.0
\][/tex]
Step 5: Write the Partial Fraction Decomposition
Using these values, the partial fraction decomposition is:
[tex]\[
\frac{2y^3 + 17y^2 + 44y + 61}{(y+5)^2(y+1)^2} = \frac{2}{y+5} + \frac{0}{y+1} + \frac{1}{(y+5)^2} + \frac{2}{(y+1)^2}
\][/tex]
Which simplifies to:
[tex]\[
\frac{2}{y+5} + \frac{1}{(y+5)^2} + \frac{2}{(y+1)^2}
\][/tex]
Step 6: Evaluate the Integral
Now, we integrate each term separately:
[tex]\[
\int \left(\frac{2}{y+5} + \frac{1}{(y+5)^2} + \frac{2}{(y+1)^2}\right) \, dy
\][/tex]
1. [tex]\(\int \frac{2}{y+5} \, dy = 2 \ln |y+5|\)[/tex]
2. [tex]\(\int \frac{1}{(y+5)^2} \, dy = \int (y+5)^{-2} \, dy = -\frac{1}{y+5}\)[/tex]
3. [tex]\(\int \frac{2}{(y+1)^2} \, dy = 2 \int (y+1)^{-2} \, dy = -\frac{2}{y+1}\)[/tex]
Combine these results:
[tex]\[
\int \frac{2y^3 + 17y^2 + 44y + 61}{(y+5)^2(y+1)^2} \, dy = 2 \ln |y+5| - \frac{1}{y+5} - \frac{2}{y+1}
\][/tex]
Finally, putting it all together, the integral of the given rational function is:
[tex]\[
2 \ln |y+5| - \frac{1}{y+5} - \frac{2}{y+1} + C
\][/tex]
where [tex]\(C\)[/tex] is the constant of integration.
Summary:
- [tex]\(A = 2.0\)[/tex]
- [tex]\(B = 0.0\)[/tex]
- [tex]\(C = 1.0\)[/tex]
- [tex]\(D = 2.0\)[/tex]
The integral of the given rational expression is:
[tex]\[
\int \frac{2y^3 + 17y^2 + 44y + 61}{(y+5)^2(y+1)^2} \, dy = 2 \ln |y+5| - \frac{1}{y+5} - \frac{2}{y+1} + C
\][/tex]
