– $ \dfrac{ x^4 \space + \space 6 }{ x^5 \space + \space 7x^3 }$
– $ \dfrac{ 2 }{ (x^2 \space – \space 9)^2 }$
The main objective of this question is to find the partial fraction decomposition for the given expressions.
This question uses the concept of partial fraction decomposition. Finding antiderivatives of several rational functions sometimes requires partial fraction decomposition. It entails factoring rational function denominators before creating a summation of fractions where denominators are indeed the factors of an original denominator.
Expert Answer
a) We are given:
\[ \frac{ x^4 \space + \space 6 }{ x^5 \space + \space 7x^3 } \]
Then:
\[ \frac{ x^4 \space + \space 6 }{ x^3 \space (x^2 \space + \space 7)} \]
Now the partial fraction is:
\[\space = \space \frac{}A{x} \space + \space \frac{B}{x^2} \space + \space {C}{x^3} \space + \space \frac{ Dx \space + \space E}{x^2 \space + \space 7 } \]
Hence, $ A, \space B, \space C, \space D, \space E $ are the constants.
The final answer is:
\[\space = \space \frac{}A{x} \space + \space \frac{B}{x^2} \space + \space {C}{x^3} \space + \space \frac{ Dx \space + \space E}{x^2 \space + \space 7 } \]
b) We are given that:
\ [\frac{ 2 }{ (x^2 \space – \space 9)^2 }\]
\[\space = \space \frac{2}{(( x \space + \space 3) \space (x \space – \space 3))^2} \]
\[\space = \space \frac{2}{( x \space + \space 3)^2 \space (x \space – \space 3)^2} \]
Now the partial fraction is:
\[\space = \space \frac{}A{x \space + \space 3} \space + \space \frac{B}{(x \space + \space 3)^2} \space + \space {C}{x \space – \space 3} \space + \space \frac{ D }{ (x \space – \space 3)^2 } \]
Hence, $ A, \space B, \space C, \space D, \space E $ are the constants.
The final answer is:
\[\space = \space \frac{}A{x \space + \space 3} \space + \space \frac{B}{(x \space + \space 3)^2} \space + \space {C}{x \space – \space 3} \space + \space \frac{ D }{ (x \space – \space 3)^2 } \]
Numerical Answer
The partial fraction decomposition for the given functions are:
\[\space = \space \frac{}A{x} \space + \space \frac{B}{x^2} \space + \space {C}{x^3} \space + \space \frac{ Dx \space + \space E}{x^2 \space + \space 7 } \]
\[\space = \space \frac{}A{x \space + \space 3} \space + \space \frac{B}{(x \space + \space 3)^2} \space + \space {C}{x \space – \space 3} \space + \space \frac{ D }{ (x \space – \space 3)^2 } \]
Example
Find the partial fraction decomposition for the given expression.
\[\frac{ x^6 \space + \space 8 }{ x^5 \space + \space 7x^3 } \]
We are given that:
\[ \frac{ x^6 \space + \space 8 }{ x^5 \space + \space 7x^3 } \]
Then:
\[ \frac{ x^6 \space + \space 8 }{ x^3 \space (x^2 \space + \space 7)} \]
Now the partial fraction is:
\[\space = \space \frac{}A{x} \space + \space \frac{B}{x^2} \space + \space {C}{x^3} \space + \space \frac{ Dx \space + \space E}{x^2 \space + \space 7 } \]
Hence, $ A, \space B, \space C, \space D, \space E $ are the constants.
The final answer is:
\[\space = \space \frac{}A{x} \space + \space \frac{B}{x^2} \space + \space {C}{x^3} \space + \space \frac{ Dx \space + \space E}{x^2 \space + \space 7 } \]