CAIE A-Level Mathematics 3.5.6 Integration by substitution

State or imply $\mathrm{d} x=\frac{1}{2} u^{-\frac{1}{2}} \mathrm{~d} u$

B1

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

M1

Obtain correct integral $\frac{1}{2} \int \mathrm{e}^{-\frac{1}{4} u} \mathrm{~d} u$

A1

OE

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

M1

u=9 and u=0 or x=3 and x=0.

Obtain answer $2-2 \mathrm{e}^{-\frac{9}{4}}$

A1

Or exact equivalent.

Alternative Method for Question 9(b)

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

M1

Recognition used.

a negative

A1

a=-2

A1

Use correct limits in an integral of the form $a \mathrm{e}^{-\frac{1}{4} x^{2}}$

M1

x=3 and x=0.

Obtain answer $2-2 \mathrm{e}^{-\frac{9}{4}}$

A1

Or exact equivalent.