6. (a) Show that $(4 + 3\sqrt{x})^2$ can be written as $16 + k\sqrt{x} + 9x$, where $k$ is a constant to be found - Edexcel - A-Level Maths Pure - Question 7 - 2007 - Paper 2

First, we expand $(4 + 3\sqrt{x})^2$ as shown in (a):
$(4 + 3\sqrt{x})^2 = 16 + 24\sqrt{x} + 9x.$

Now, we find the integral:

$\int (4 + 3\sqrt{x})^2 dx = \int (16 + 24\sqrt{x} + 9x) dx.$

We integrate each term separately:

• The integral of $16$ is $16x$.
• The integral of $24\sqrt{x}$ is $24 \cdot \frac{2}{3} x^{3/2} = 16 x^{3/2}.$
• The integral of $9x$ is $\frac{9}{2} x^2$.

Combining all these results, we have:

$\int (4 + 3\sqrt{x})^2 dx = 16x + 16x^{3/2} + \frac{9}{2}x^2 + C,$
where $C$ is the constant of integration.