CAIE A-Level Mathematics AS 1.8 Integration Questions

Integrate to obtain form $k(5 x-3)^{-1}$

*M1

OE

$4(5 x-3)^{-1}$
Or unsimplified equivalent.
Condone absence of $\ldots+c$ so far.

Substitute $x=\frac{4}{5}$ and y=-3 to attempt value of c
DM0 for substituting $\left(-3, \frac{4}{5}\right)$.

$y=4(5 x-3)^{-1}-7$ allow f(x) or $f=4(5 x-3)^{-1}-7$
OE
Condone c=-7 as the final answer providing
y= or $f(x)=\frac{4}{(5 x-3)}+c$ OE is seen earlier.
Attempts to write equation in y=m x+c form
scores A0. Do not ISW. Gains max 3/4.

Integrate to obtain $-2 x^{-1}$
OE

Substitute limits correctly with clear indication seen that upper limit gives 0
For integral of form $-k x^{-n}$, where k>0, n>0.

Obtain $\frac{2}{3}$
WWW

Marking guidance:

Accept 0.667.

State or imply volume is $\pi \int\left(2 x^{3}+10\right) \mathrm{d} x$
Implied if $\pi$ appears only at the end.

Marking guidance:

Do not allow an unsimplified:
$$\pi \int\left(\left(2 x^{3}+10\right)^{1 / 2}\right)^{2}$$

Integrate to obtain $k_{1} x^{4}+k_{2} x$ and evaluate using limits 1 and 3
Where $k_{1} k_{2} \neq 0$.

$60 \pi$
OE

Allow from a correct integral and sight of limits. Allow numerical answers in the range 188-189.