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4. (a) Find the values of the constants A, B and C - Edexcel - A-Level Maths: Pure - Question 6 - 2007 - Paper 7
Question 6
4. (a) Find the values of the constants A, B and C. (b) Hence show that the exact value of $$\int_{0}^{2} \frac{2(4x^2+1)}{(2x+1)(2x-1)} \ dx$$ is $2 + \ln k$, gi... show full transcript
Worked Solution & Example Answer:4. (a) Find the values of the constants A, B and C - Edexcel - A-Level Maths: Pure - Question 6 - 2007 - Paper 7
Step 1
Find the values of the constants A, B and C.
Answer
To find constants A, B, and C, we can use partial fraction decomposition on the expression:
By multiplying both sides by the denominator ((2x + 1)(2x - 1)), we have:
Expanding the right-hand side gives:
Now, regrouping terms:
Matching coefficients:
- For : No contribution, so this doesn't give useful information.
- For : , leading to . Thus, .
- For the constant: .
Substituting , we get:
We can select , so:
- .
Step 2
Hence show that the exact value of $$\int_{0}^{2} \frac{2(4x^2 + 1)}{(2x + 1)(2x - 1)} \ dx$$ is $2 + \ln k$, giving the value of the constant k.
Answer
The integral can be expressed as:
Using our results from part (a) and substituting into the integral leads to:
To solve:
-
Integrate each term separately:
- requires logarithmic substitution.
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By applying limits from 0 to 2 and simplifying:
After completing the calculations, we arrive at:
To find the constant k, equate and simplify the results obtained from the limits, potentially yielding: