Answer :
To find the integral of the function [tex]\( F(s) = \frac{6s^2 + 62s + 92}{(s+1)(s^2 + 10s + 51)} \)[/tex], we need to perform partial fraction decomposition and then integrate each component separately. Here’s a step-by-step guide to solving this integral:
1. Partial Fraction Decomposition:
The function [tex]\( F(s) \)[/tex] is already a rational expression where the degree of the numerator is less than the degree of the denominator, which is suitable for partial fraction decomposition.
The denominator can be factored as [tex]\( (s + 1)(s^2 + 10s + 51) \)[/tex]. For partial fraction decomposition, we assume:
[tex]\[
\frac{6s^2 + 62s + 92}{(s+1)(s^2 + 10s + 51)} = \frac{A}{s+1} + \frac{Bs + C}{s^2 + 10s + 51}
\][/tex]
2. Determine Coefficients [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex]:
Multiply both sides by the denominator to clear the fractions:
[tex]\[
6s^2 + 62s + 92 = A(s^2 + 10s + 51) + (Bs + C)(s + 1)
\][/tex]
By expanding and equating coefficients or substituting convenient values for [tex]\( s \)[/tex], solve for [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex].
3. Integration:
With the coefficients found, we now integrate each part. Separate the integral:
[tex]\[
\int \left(\frac{A}{s+1} + \frac{Bs + C}{s^2 + 10s + 51}\right) ds = \int \frac{A}{s+1} ds + \int \frac{Bs + C}{s^2 + 10s + 51} ds
\][/tex]
- The integral [tex]\(\int \frac{A}{s+1} ds\)[/tex] is straightforward:
[tex]\[
A \cdot \int \frac{1}{s+1} ds = A \cdot \ln|s+1|
\][/tex]
- For the integral [tex]\(\int \frac{Bs + C}{s^2 + 10s + 51} ds\)[/tex], complete the square for the quadratic expression in the denominator:
[tex]\[
s^2 + 10s + 51 = (s + 5)^2 + 26
\][/tex]
Therefore, you can separate the integral into:
[tex]\[
\int \frac{Bs + C}{(s + 5)^2 + 26} ds = B \cdot \int \frac{s}{(s + 5)^2 + 26} ds + C \cdot \int \frac{1}{(s + 5)^2 + 26} ds
\][/tex]
- For [tex]\( B \cdot \int \frac{s}{(s+5)^2+26} ds \)[/tex], a trigonometric substitution or recognizing the derivative form will work:
[tex]\[
\int \frac{s}{(s+5)^2+26} ds
\][/tex]
- For [tex]\( C \cdot \int \frac{1}{(s + 5)^2 + 26} ds\)[/tex], use the arctangent formula:
[tex]\[
C \cdot \int \frac{1}{(s + 5)^2 + 26} ds = \frac{C}{\sqrt{26}} \cdot \arctan\left(\frac{s+5}{\sqrt{26}}\right)
\][/tex]
4. Combine Results:
After evaluating each integral, combine them to form the antiderivative [tex]\( \int F(s) ds \)[/tex].
The calculated result of the integral is:
[tex]\[
\frac{6}{7} \ln|s+1| + \frac{18}{7} \ln|s^2 + 10s + 51| + \frac{79\sqrt{26}}{91} \arctan\left(\frac{\sqrt{26}s}{26} + \frac{5\sqrt{26}}{26}\right)
\][/tex]
Feel free to ask any questions if you need more details on any part of the process!
1. Partial Fraction Decomposition:
The function [tex]\( F(s) \)[/tex] is already a rational expression where the degree of the numerator is less than the degree of the denominator, which is suitable for partial fraction decomposition.
The denominator can be factored as [tex]\( (s + 1)(s^2 + 10s + 51) \)[/tex]. For partial fraction decomposition, we assume:
[tex]\[
\frac{6s^2 + 62s + 92}{(s+1)(s^2 + 10s + 51)} = \frac{A}{s+1} + \frac{Bs + C}{s^2 + 10s + 51}
\][/tex]
2. Determine Coefficients [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex]:
Multiply both sides by the denominator to clear the fractions:
[tex]\[
6s^2 + 62s + 92 = A(s^2 + 10s + 51) + (Bs + C)(s + 1)
\][/tex]
By expanding and equating coefficients or substituting convenient values for [tex]\( s \)[/tex], solve for [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex].
3. Integration:
With the coefficients found, we now integrate each part. Separate the integral:
[tex]\[
\int \left(\frac{A}{s+1} + \frac{Bs + C}{s^2 + 10s + 51}\right) ds = \int \frac{A}{s+1} ds + \int \frac{Bs + C}{s^2 + 10s + 51} ds
\][/tex]
- The integral [tex]\(\int \frac{A}{s+1} ds\)[/tex] is straightforward:
[tex]\[
A \cdot \int \frac{1}{s+1} ds = A \cdot \ln|s+1|
\][/tex]
- For the integral [tex]\(\int \frac{Bs + C}{s^2 + 10s + 51} ds\)[/tex], complete the square for the quadratic expression in the denominator:
[tex]\[
s^2 + 10s + 51 = (s + 5)^2 + 26
\][/tex]
Therefore, you can separate the integral into:
[tex]\[
\int \frac{Bs + C}{(s + 5)^2 + 26} ds = B \cdot \int \frac{s}{(s + 5)^2 + 26} ds + C \cdot \int \frac{1}{(s + 5)^2 + 26} ds
\][/tex]
- For [tex]\( B \cdot \int \frac{s}{(s+5)^2+26} ds \)[/tex], a trigonometric substitution or recognizing the derivative form will work:
[tex]\[
\int \frac{s}{(s+5)^2+26} ds
\][/tex]
- For [tex]\( C \cdot \int \frac{1}{(s + 5)^2 + 26} ds\)[/tex], use the arctangent formula:
[tex]\[
C \cdot \int \frac{1}{(s + 5)^2 + 26} ds = \frac{C}{\sqrt{26}} \cdot \arctan\left(\frac{s+5}{\sqrt{26}}\right)
\][/tex]
4. Combine Results:
After evaluating each integral, combine them to form the antiderivative [tex]\( \int F(s) ds \)[/tex].
The calculated result of the integral is:
[tex]\[
\frac{6}{7} \ln|s+1| + \frac{18}{7} \ln|s^2 + 10s + 51| + \frac{79\sqrt{26}}{91} \arctan\left(\frac{\sqrt{26}s}{26} + \frac{5\sqrt{26}}{26}\right)
\][/tex]
Feel free to ask any questions if you need more details on any part of the process!
