Figure 2 shows a sketch of part of the curve with equation y = (lnx)² x > 0 The finite region R, shown shaded in Figure 2, is bounded by the curve, the line with equation x = 2, the x-axis and the line with equation x = 4 - Edexcel - A-Level Maths Pure - Question 13 - 2021 - Paper 1

To find the exact area of R, we need to integrate the curve from x = 2 to x = 4:

$ext{Area} = \int_{2}^{4} (\ln x)^{2} \, dx$

Using integration by parts, let:

• u = (ln x)² → dv = dx
• du = 2(ln x)(1/x) dx → v = x

Now applying integration by parts:

$\int u \, dv = uv - \int v \, du$

Thus,

$A = \left[ x(\ln x)^{2} \right]_{2}^{4} - \int_{2}^{4} x \cdot 2 \frac{\ln x}{x} \, dx$

This simplifies to:

$= \left[ 4(\ln 4)^{2} - 2(\ln 2)^{2} \right] - 2 \int (\ln x) \, dx$

The integral of ( \ln x ) is given by:

$\int \ln x \, dx = x \ln x - x + C$

Evaluating the limits and substituting back yields:

After calculations, the final area can be expressed in the form ( y = a(\ln 2)^{2} + b\ln 2 + c ) where we need to find integer values of a, b, and c.