A-Level Edexcel Maths Pure Vectors in 3 Dimensions: 4. (a) Express \( \frac{25}{x^2(

4. (a) Express $ \frac{25}{x^2(2x+1)} $ in partial fractions - Edexcel - A-Level Maths Pure - Question 6 - 2014 - Paper 8

Question 6

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4. (a) Express $ \frac{25}{x^2(2x+1)} $ in partial fractions. (b) Use calculus to find the exact volume of the solid of revolution generated, giving your answer... show full transcript

Worked Solution & Example Answer:4. (a) Express $ \frac{25}{x^2(2x+1)} $ in partial fractions - Edexcel - A-Level Maths Pure - Question 6 - 2014 - Paper 8

Step 1

Express $ \frac{25}{x^2(2x+1)} $ in partial fractions

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Answer

To express ( \frac{25}{x^2(2x+1)} ) in partial fractions, we assume it can be decomposed as follows:

$\frac{25}{x^2(2x+1)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{2x+1}$

Multiplying both sides by ( x^2(2x+1) ) gives:

$25 = A x(2x+1) + B(2x+1) + C x^2$

Setting up equations by substituting suitable values for ( x ):

Let ( x = 0 ):
$25 = B(1) \Rightarrow B = 25$
Let ( x = 1 ):
$25 = A(1)(3) + 25(3) + C(1) \Rightarrow 25 = 3A + 75 + C \Rightarrow C = -50 - 3A$
For the coefficient of ( x^2 ) in ( 25 = (A + C)x^2 + (2B + 2C)x + (B) ):
$0 = A + C\ ext{(coefficient of } x^2\text{)} \Rightarrow 0 = A - 50 - 3A \Rightarrow 2A = 50 \Rightarrow A = 25$

Substituting ( A = 25 ) back gives ( C = -50 - 75 \Rightarrow C = -125 ).

Thus, we get:

$\frac{25}{x^2(2x+1)} = \frac{25}{x} + \frac{25}{x^2} - \frac{125}{2x+1}$

Step 2

Use calculus to find the exact volume of the solid of revolution generated

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Answer

To find the volume ( V ) of the solid of revolution generated when region ( R ) is rotated around the x-axis, we use the formula:

$V = \int_{1}^{4} \pi [f(x)]^2 \, dx$

In this case, ( f(x) = \frac{5}{\sqrt{2x+1}} ) so:

$V = \pi \int_{1}^{4} \left(\frac{5}{\sqrt{2x+1}}\right)^2 \, dx = \pi \int_{1}^{4} \frac{25}{2x+1} \, dx$

Evaluating the integral:

$V = \pi \cdot \left[ 25 \ln(2x+1) \right]_{1}^{4}$

This simplifies to:

$V = \pi \left( 25 \ln(9) - 25 \ln(3) \right) = 25\pi \ln(3)$

Finally, we represent this in the required form:

$V = 25\pi(\ln(3) - \ln(1)) + 25\pi$

So, letting ( a = 25\pi, b = 25\pi, c = 3 ) gives us:

$V = a + b\ln(c)$

4. (a) Express \( \frac{25}{x^2(2x+1)} \) in partial fractions - Edexcel - A-Level Maths Pure - Question 6 - 2014 - Paper 8

Worked Solution & Example Answer:4. (a) Express \( \frac{25}{x^2(2x+1)} \) in partial fractions - Edexcel - A-Level Maths Pure - Question 6 - 2014 - Paper 8

Express \( \frac{25}{x^2(2x+1)} \) in partial fractions

Use calculus to find the exact volume of the solid of revolution generated

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