4. (a) Express $$\frac{25}{x^2(2x + 1)}$$ in partial fractions - Edexcel - A-Level Maths Pure - Question 7 - 2014 - Paper 8

The volume (V) of the solid generated by rotating region (R) around the x-axis can be computed using the formula:

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

Here, the function defined by the curve (C) is:

$f(x) = \frac{5}{\sqrt{2x + 1}}$

Thus, we have:

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

This simplifies to:

$V = \pi \int_{1}^{4} \frac{25}{2x + 1} \,dx$

Computing the integral, we can separate the terms:

$V = 25\pi \int_{1}^{4} \frac{1}{2x + 1} \,dx$

Using the substitution (u = 2x + 1) yields:

$du = 2 \,dx \Rightarrow dx = \frac{du}{2}$

Changing the limits accordingly, when (x=1), (u=3) and when (x=4), (u=9). Thus, the entry becomes:

$V = \frac{25\pi}{2} \int_{3}^{9} \frac{1}{u} \,du$

Evaluating the integral gives:

$V = \frac{25\pi}{2} \left[\ln|u|\right]_{3}^{9} = \frac{25\pi}{2} (\ln(9) - \ln(3)) = \frac{25\pi}{2} \ln(3)$

Finally, we can express the volume in the required form:

$\frac{75\pi}{2} \ell(\frac{3}{1})$
where:
a = 0, b = \frac{75\pi}{2}, c = 3.