Express \({x^2} + 3x + 2\) in the form \({(x + h)^2} + k\).

Factorize \({x^2} + 3x + 2\).

Sketch the graph of \(f(x)\), indicating on it the equations of the asymptotes, the coordinates of the \(y\)-intercept and the local maximum.

Show that \(\frac{1}{{x + 1}} - \frac{1}{{x + 2}} = \frac{1}{{{x^2} + 3x + 2}}\).

Hence find the value of \(p\) if \(\int_0^1 {f(x){\text{d}}x = \ln (p)} \).

Sketch the graph of \(y = f\left( {\left| x \right|} \right)\).

Determine the area of the region enclosed between the graph of \(y = f\left( {\left| x \right|} \right)\), the \(x\)-axis and the lines with equations \(x = - 1\) and \(x = 1\).

\({x^2} + 3x + 2 = {\left( {x + \frac{3}{2}} \right)^2} - \frac{1}{4}\)     A1

[1 mark]

\({x^2} + 3x + 2 = (x + 2)(x + 1)\)     A1

[1 mark]

A1 for the shape

A1 for the equation \(y = 0\)

A1 for asymptotes \(x = - 2\) and \(x = - 1\)

A1 for coordinates \(\left( { - \frac{3}{2},{\text{ }} - 4} \right)\)

A1 \(y\)-intercept \(\left( {0,{\text{ }}\frac{1}{2}} \right)\)

[5 marks]

\(\frac{1}{{x + 1}} - \frac{1}{{x + 2}} = \frac{{(x + 2) - (x + 1)}}{{(x + 1)(x + 2)}}\)     M1

\( = \frac{1}{{{x^2} + 3x + 2}}\)     AG

[1 mark]

\(\int\limits_0^1 {\frac{1}{{x + 1}} - \frac{1}{{x + 2}}{\text{d}}x} \)

\( = \left[ {\ln (x + 1) - \ln (x + 2)} \right]_0^1\)     A1

\( = \ln 2 - \ln 3 - \ln 1 + \ln 2\)     M1

\( = \ln \left( {\frac{4}{3}} \right)\)     M1A1

\(\therefore p = \frac{4}{3}\)

[4 marks]

symmetry about the \(y\)-axis     M1

correct shape     A1

Note:     Allow FT from part (b).

[2 marks]

\(2\int_0^1 {f(x){\text{d}}x} \)     (M1)(A1)

\( = 2\ln \left( {\frac{4}{3}} \right)\)     A1

Note:     Do not award FT from part (e).

[3 marks]