CAIE A-Level Mathematics A2 3.5.5 Inverse Tangent Questions

Use integration by parts to obtain $p x \mathrm{e}^{-a x}+q \int \mathrm{e}^{-a x} \mathrm{~d} x$

*M1

Condone sign error in parts formula and omission of d x.
M0 if a value is used for a (not equivalent work).

Obtain $-\frac{1}{a} x \mathrm{e}^{-a x}+\frac{1}{a} \int \mathrm{e}^{-a x} \mathrm{~d} x$
OE

Complete integration to obtain $-\frac{1}{a} x \mathrm{e}^{-a x}-\frac{1}{a^{2}} \mathrm{e}^{-a x}$
OE

Correct use of limits 0 and $\frac{2}{a}$ in an expression of the form
$r x \mathrm{e}^{-a x}+s \mathrm{e}^{-a x}$$\left(\frac{-2}{a^{2}} \mathrm{e}^{-2}-\frac{1}{a^{2}} \mathrm{e}^{-2}+0+\frac{1}{a^{2}}\right)$

Obtain $\frac{1}{a^{2}}\left(1-3 \mathrm{e}^{-2}\right)$
ISW

Or simplified 2-term equivalent, e.g. $\frac{\mathrm{e}^{2}-3}{a^{2} \mathrm{e}^{2}}$.