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3. (a) Express \( \frac{5}{(x-1)(3x+2)} \) in partial fractions - Edexcel - A-Level Maths Pure - Question 7 - 2011 - Paper 6
Question 7
3. (a) Express \( \frac{5}{(x-1)(3x+2)} \) in partial fractions. (b) Hence find \( \int \frac{5}{(x-1)(3x+2)} \, dx \), where \( x > 1 \). (c) Find the particu... show full transcript
Worked Solution & Example Answer:3. (a) Express \( \frac{5}{(x-1)(3x+2)} \) in partial fractions - Edexcel - A-Level Maths Pure - Question 7 - 2011 - Paper 6
Step 1
Express \( \frac{5}{(x-1)(3x+2)} \) in partial fractions.
Answer
To express ( \frac{5}{(x-1)(3x+2)} ) in partial fractions, we set up the equation:
[ \frac{5}{(x-1)(3x+2)} = \frac{A}{x-1} + \frac{B}{3x+2} ]
Multiplying through by ( (x-1)(3x+2) ) gives:
[ 5 = A(3x+2) + B(x-1) ]
Substituting convenient values of ( x ):
- Let ( x = 1 ): [ 5 = A(3(1)+2) \Rightarrow 5 = 5A \Rightarrow A = 1 ]
- Let ( x = -\frac{2}{3} ): [ 5 = B(-\frac{2}{3}-1) \Rightarrow 5 = B(-\frac{5}{3}) \Rightarrow B = -3 ]
Thus, the partial fraction decomposition is:
[ \frac{5}{(x-1)(3x+2)} = \frac{1}{x-1} - \frac{3}{3x+2} ]
Step 2
Hence find \( \int \frac{5}{(x-1)(3x+2)} \, dx \), where \( x > 1 \).
Answer
Using the partial fraction result, we can integrate:
[ \int \frac{5}{(x-1)(3x+2)} , dx = \int \left( \frac{1}{x-1} - \frac{3}{3x+2} \right) , dx ]
This results in:
[ = \ln |x-1| - \ln |3x+2| + C = \ln \left( \frac{x-1}{3x+2} \right) + C ]
Step 3
Find the particular solution of the differential equation \( (x-1)(3x+2) \frac{dy}{dx} = 5y \), \( x > 1 \), for which \( y=8 \) at \( x=2 \).
Answer
We first rewrite the equation:
[ \frac{dy}{y} = \frac{5}{(x-1)(3x+2)} , dx ]
Integrating both sides:
[ \ln |y| = \int \frac{5}{(x-1)(3x+2)} , dx ]
Substituting our previous solution:
[ = \ln \left( \frac{x-1}{3x+2} \right) + C ]
Exponentiating both sides gives:
[ y = K \cdot \frac{x-1}{3x+2} ]
Using the condition ( y = 8 ) at ( x = 2 ):
[ 8 = K \cdot \frac{2-1}{3(2)+2} \Rightarrow 8 = K \cdot \frac{1}{8} \Rightarrow K = 64 ]
Thus, the particular solution is:
[ y = \frac{64(x-1)}{3x+2} ]