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3. (a) Express \( \frac{5x + 3}{(2x - 3)(x + 2)} \) in partial fractions - Edexcel - A-Level Maths Pure - Question 5 - 2005 - Paper 6
Question 5
3. (a) Express \( \frac{5x + 3}{(2x - 3)(x + 2)} \) in partial fractions. (b) Hence find the exact value of \( \int \frac{5x + 3}{(2x - 3)(x + 2)} dx \), giving you... show full transcript
Worked Solution & Example Answer:3. (a) Express \( \frac{5x + 3}{(2x - 3)(x + 2)} \) in partial fractions - Edexcel - A-Level Maths Pure - Question 5 - 2005 - Paper 6
Step 1
Express \( \frac{5x + 3}{(2x - 3)(x + 2)} \) in partial fractions.
Answer
To express ( \frac{5x + 3}{(2x - 3)(x + 2)} ) in partial fractions, we assume:
[ \frac{5x + 3}{(2x - 3)(x + 2)} = \frac{A}{2x - 3} + \frac{B}{x + 2} ]
Multiplying through by the denominator ( (2x - 3)(x + 2) ) gives:
[ 5x + 3 = A(x + 2) + B(2x - 3) ]
Next, we can substitute convenient values to find ( A ) and ( B ).
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Let ( x = \frac{3}{2} ):
[ 5\left(\frac{3}{2}\right) + 3 = A\left(\frac{3}{2} + 2\right) + 0\Rightarrow A = 3 ] -
Let ( x = -2 ):
[ 5(-2) + 3 = 0 + B(2(-2) - 3) \Rightarrow B = -1 ]
Thus, we have:
[ \frac{5x + 3}{(2x - 3)(x + 2)} = \frac{3}{2x - 3} - \frac{1}{x + 2} ]
Step 2
Hence find the exact value of \( \int \frac{5x + 3}{(2x - 3)(x + 2)} dx \), giving your answer as a single logarithm.
Answer
Using the partial fractions found previously, we can evaluate the integral:
[ \int \frac{5x + 3}{(2x - 3)(x + 2)} dx = \int \left( \frac{3}{2x - 3} - \frac{1}{x + 2} \right) dx ]
This leads to:
[ = 3 \ln |2x - 3| - \ln |x + 2| + C ]
Combining the logarithms:
[ = \ln |(2x - 3)^3| - \ln |x + 2| = \ln \left| \frac{(2x - 3)^3}{x + 2} \right| + C ]
Thus, we simplify further for the evaluated definite integral.
The result of the integral from ( \int \frac{5x + 3}{(2x - 3)(x + 2)} dx ) will yield a final answer as a single logarithm.