CAIE A-Level Mathematics A2 3.5 Integration Questions

Split fraction to obtain $1+\frac{x-4}{x^{2}+4}$
Attempt integration and obtain $p \ln \left(x^{2}+4\right)$ or $q \tan ^{-1}\left(\frac{x}{2}\right)$ from correct
working
Marking guidance:

Allow for $p \ln \left(x^{2}+4\right)$ from $\int \frac{x}{x^{2}+4} \mathrm{~d} x$ but only if a
correct method for splitting has been used.

Obtain $\frac{1}{2} \ln \left(x^{2}+4\right)$

A1 FT

Follow through is on their coefficients in the partial fraction.

Allow from $\frac{x^{2}}{x^{2}+4}+\frac{x}{x^{2}+4}$ even if the split of the fraction is not complete. If $1-\frac{4}{x^{2}+4}+\frac{x}{x^{2}+4}$ later seen or implied, award the B1.

Only available from a correct split, not from an approach using parts that is incomplete.

Obtain $-2 \tan ^{-1}\left(\frac{x}{2}\right)$

A1 FT

Only available from a correct split, not from an approach using parts that is incomplete.

Correct use of correct limits 0 and 6 in an expression involving $p \ln \left(x^{2}+4\right)$,
$q \tan ^{-1}\left(\frac{x}{2}\right)$ and no incorrect terms.
p and q should be constants.

The x term is not required at this stage.

Obtain $6+\frac{1}{2} \ln 10-2 \tan ^{-1} 3$
ISW

Or three term equivalent. (Must combine the ln terms.) Accept with $\frac{1}{2} \ln |10|$.

5

Alternative method for question 5

Use the substitution $x=2 \tan \theta$ to obtain $\int 2 \tan ^{2} \theta+\tan \theta \mathrm{d} \theta$
Attempt integration and obtain $p \tan \theta$ or $r \ln (\sec \theta)$ from correct working
Obtain $2 \tan \theta(-2 \theta)$ and

A1 FT

Follow through on their coefficients after the substitution.

Obtain ln $\sec \theta$

A1 FT

Follow through on their coefficients after the substitution.

Use correct limits 0 and $\tan ^{-1} 3$ in an expression involving $u \tan \theta, v \ln \sec \theta$
and no incorrect terms
u and v should be constants. The $\theta$ term is not required at
this stage.

Obtain $6+\ln \left|\sec \left(\tan ^{-1} 3\right)\right|-2 \tan ^{-1} 3$
ISW

Or three term equivalent.

Not required to simplify $\ln \left|\sec \left(\tan ^{-1} 3\right)\right|$.

B1 M1 A1ft A1ft M1 A1

Use part (a) and correct double angle formula to obtain expression
involving $\int \sin ^{2} 2 \theta \mathrm{~d} \theta$ or $\int \cos ^{2} 2 \theta \mathrm{~d} \theta$$\int \cos ^{4} \theta-\sin ^{4} \theta+4 \sin ^{2} \theta \cos ^{2} \theta \mathrm{~d} \theta=\int \cos 2 \theta+\sin ^{2} 2 \theta \mathrm{~d} \theta$

Marking guidance:

Allow BOD for $2 \sin ^{2} 2 \theta$ if $\sin 2 \theta=2 \sin \theta \cos \theta$ seen.

$\int \cos 2 \theta d \theta=\frac{1}{2} \sin 2 \theta$
Seen or implied.

Use of correct double angle formula on second part of the integral to obtain a form that can be integrated directly
e.g. $\int \sin ^{2} 2 \theta \mathrm{~d} \theta=\int \frac{1-\cos 4 \theta}{2} \mathrm{~d} \theta$

Obtain $\frac{1}{2} \theta-\frac{1}{8} \sin 4 \theta$
Condone a mixture of x and $\theta$.

Correct use of limits $\pm \frac{\pi}{8}$ in an expression of the form
$p \theta+q \sin 2 \theta+r \sin 4 \theta$ and evaluate the trig
$\left(2\left(\frac{1}{2} \times \frac{1}{\sqrt{2}}+\frac{\pi}{16}-\frac{1}{8}\right)\right)$

Obtain $\frac{1}{2} \sqrt{2}+\frac{1}{8} \pi-\frac{1}{4}$
ISW
Or exact equivalent from exact working.

Separate variables correctly
$\int \frac{1}{x(300-x)} \mathrm{d} x=\int \frac{1}{1495} \mathrm{~d} t$

Correct integration of t term
E.g. obtain t or $\frac{t}{1495}$.

State or imply partial fractions of the form $\frac{A}{x}+\frac{B}{300-x}$
Correct method to find A or B
$A=\frac{1}{300}$ and $B=\frac{1}{300}$.

May see $A=B=\frac{1495}{300}=\frac{299}{60}$.

Obtain terms $\frac{1495}{300} \ln x-\frac{1495}{300} \ln (300-x)$
OE. May see $\frac{1}{300} \ln x-\frac{1}{300} \ln (300-x)$.

Use t=0, x=1 to evaluate a constant or as limits in a solution containing terms
of the form $\ln x, \ln (300-x)$ and t.
Obtain correct answer in any form
E.g. $\frac{1495}{300}[\ln x-\ln (300-x)]=t-\frac{1495}{300} \ln 299$.

Use law of logarithms twice to obtain an expression for t

М1

Obtain final answer $t=\frac{299}{60} \ln \frac{299 x}{300-x}$ or equivalent single logarithm

State integral of the form $\mp \ln \cos \theta$ or $\pm \ln \sec \theta$
*M1
$[-\ln \cos \theta]_{\frac{\pi}{4}}^{\frac{\pi}{3}} \mathrm{OE}$
Use correct limits correctly and insert exact values for the trigonometric
ratios
Need to see evidence of the substitution
Obtain a correct expression, e.g. $-\ln \frac{1}{2}+\ln \frac{1}{\sqrt{2}}$
Obtain $\frac{1}{2} \ln 2$ from correct working
AG (must see an intermediate step)

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. $\frac{\mathrm{e}^{2}-3}{a^{2} \mathrm{e}^{2}}$.

State or imply $\mathrm{d} u=-\cos x \mathrm{~d} x$
Use $\sin 2 x=2 \sin x \cos x$ and write the integral in terms of u

*M1

Obtain $\pm 2 \int \frac{(1-u)}{\sqrt{u}} \mathrm{~d} u$ or equivalent
Integrate correctly to obtain $a u^{\frac{1}{2}}+b u^{\frac{3}{2}}$
Obtain correct $-4 u^{\frac{1}{2}}+\frac{4}{3} u^{\frac{3}{2}}$
Correctly use limits u=2 and 0 in an expression of the form $a u^{\frac{1}{2}}+b u^{\frac{3}{2}}$ OR limits $x=\frac{3}{2} \pi$ and $\frac{1}{2} \pi$ in an expression of the form
$$a(1-\sin x)^{\frac{1}{2}}+b(1-\sin x)^{\frac{3}{2}}$$
Obtain $\frac{8}{3}-\frac{4}{3} \sqrt{2}$