4. (a) Express $$\lim_{\alpha \to 0} \sum_{x=2.1}^{6.3} \delta x \cdot \frac{2}{x}$$ as an integral - Edexcel - A-Level Maths Pure - Question 4 - 2022 - Paper 1

From the previous part, we have:

$\int_{2.1}^{6.3} \frac{2}{x} \, dx.$

Now, we calculate the integral:

$\int \frac{2}{x} \, dx = 2 \ln |x| + C,$

Thus, we can evaluate the definite integral:

$2 \ln(x) \Big|_{2.1}^{6.3} = 2 \ln(6.3) - 2 \ln(2.1) = 2 [\ln(6.3) - \ln(2.1)] = 2 \ln\left( \frac{6.3}{2.1} \right).$

We set this equal to $\ln k$, which simplifies to:

$\ln\left( k \right) = 2 \ln\left( 3 \right)$ leading to:

$k = e^{2 \ln 3} = 3^2 = 9.$

Thus, we have shown that:

$\lim_{\alpha \to 0} \sum_{x=2.1}^{6.3} \delta x \cdot \frac{2}{x} = \ln 9.$