CAIE A-Level Mathematics A2 3.5.6 Integration By Substitution Questions

State or imply $\mathrm{d} x=\frac{1}{2} u^{-\frac{1}{2}} \mathrm{~d} u$
Or equivalent e.g. $\mathrm{d} u=2 x \mathrm{~d} x$.

Alternative substitution: $u=-\frac{1}{4} x^{2}$.

Substitute for x and d x
Obtain correct integral $\frac{1}{2} \int \mathrm{e}^{-\frac{1}{4} u} \mathrm{~d} u$
OE

Use correct limits in an integral of the form $a \mathrm{e}^{-\frac{1}{4} u}$ or $a \mathrm{e}^{-\frac{1}{4} x^{2}}$
u=9 and u=0 or x=3 and x=0.

Obtain answer $2-2 \mathrm{e}^{-\frac{9}{4}}$
Or exact equivalent.

Alternative Method for Question 9(b)

$\int x \mathrm{e}^{-\frac{1}{4} x^{2}} \mathrm{~d} x=a \mathrm{e}^{-\frac{1}{4} x^{2}}$
Recognition used.

a negative
a=-2
Use correct limits in an integral of the form $a \mathrm{e}^{-\frac{1}{4} x^{2}}$
x=3 and x=0.

Obtain answer $2-2 \mathrm{e}^{-\frac{9}{4}}$
Or exact equivalent.