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a) Express \( \frac{25}{x^2(2x + 1)} \) in partial fractions - Edexcel - A-Level Maths Pure - Question 4 - 2014 - Paper 8
Question 4
a) Express \( \frac{25}{x^2(2x + 1)} \) in partial fractions. b) (Figure 2 shows a sketch of part of the curve \( C \) with equation \( y = \frac{5}{\sqrt{(2 + x)... show full transcript
Worked Solution & Example Answer:a) Express \( \frac{25}{x^2(2x + 1)} \) in partial fractions - Edexcel - A-Level Maths Pure - Question 4 - 2014 - Paper 8
Step 1
Express \( \frac{25}{x^2(2x + 1)} \) in partial fractions
Answer
To express ( \frac{25}{x^2(2x + 1)} ) in partial fractions, we can start by assuming a form:
[ \frac{25}{x^2(2x + 1)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{2x + 1} ]
Multiplying both sides by the denominator ( x^2(2x + 1) ) yields:
[ 25 = A x(2x + 1) + B(2x + 1) + C x^2 ]
Expanding this gives:
[ 25 = A (2x^2 + x) + B(2x + 1) + C x^2 ]
Thus, rearranging terms, we find:
[ 25 = (2A + C) x^2 + (A + 2B) x + B ]
To find constants ( A, B, C ), we equate coefficients:
- For ( x^2 ): ( 2A + C = 0 )
- For ( x ): ( A + 2B = 0 )
- Constant term: ( B = 25 )
From equation 3, substituting ( B = 25 ) into equation 2 yields:
[ A + 50 = 0 \Rightarrow A = -50 ]
Substituting ( A = -50 ) into equation 1:
[ 2(-50) + C = 0 \Rightarrow C = 100 ]
Thus, we have:
[
\frac{25}{x^2(2x + 1)} = \frac{-50}{x} + \frac{25}{x^2} + \frac{100}{2x + 1}
]
Step 2
Use calculus to find the exact volume of the solid of revolution generated
Answer
To find the volume of the solid of revolution generated by rotating region ( R ) about the x-axis:
[ V = \int_{1}^{4} \pi y^2 , dx ]
Substituting ( y = \frac{5}{\sqrt{(2 + x)}} ), we have:
[ V = \int_{1}^{4} \pi \left( \frac{5}{\sqrt{(2 + x)}} \right)^2 , dx = \int_{1}^{4} \pi \frac{25}{2 + x} , dx ]
This integral can be computed as follows:
[ V = 25\pi\int_{1}^{4} \frac{1}{2 + x} , dx = 25\pi [\ln(2 + x)]_{1}^{4} ]
Evaluating the integral results in:
[ V = 25\pi [\ln(6) - \ln(3)] = 25\pi \ln\left( \frac{6}{3} \right) = 25\pi \ln(2) ]
Therefore, the exact volume of the solid of revolution is:
[ V = 25\pi \ln(2) ]