CAIE A-Level Mathematics 3.5 Integration Question Bank

Split fraction to obtain $1+\frac{x-4}{x^{2}+4}$

B1

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

M1

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.

M1

p and q should be constants.

The x term is not required at this stage.

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

A1

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 $\int 2 \tan ^{2} \theta+\tan \theta \mathrm{d} \theta$

B1

Attempt integration and obtain $p \tan \theta$ or $r \ln (\sec \theta)$ from correct working

M1

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

M1

u and v should be constants. The $\theta$ term is not required at
this stage.

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

A1

ISW

Or three term equivalent.

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

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

M1

$\int \cos ^{4} \theta-\sin ^{4} \theta+4 \sin ^{2} \theta \cos ^{2} \theta \mathrm{~d} \theta=\int \cos 2 \theta+\sin ^{2} 2 \theta \mathrm{~d} \theta$

Marking guidance:

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

$\int \cos 2 \theta d \theta=\frac{1}{2} \sin 2 \theta$

B1

Seen or implied.

Use of correct double angle formula on second part of the integral to obtain a form that can be integrated directly

M1

e.g. $\int \sin ^{2} 2 \theta \mathrm{~d} \theta=\int \frac{1-\cos 4 \theta}{2} \mathrm{~d} \theta$

Obtain $\frac{1}{2} \theta-\frac{1}{8} \sin 4 \theta$

A1

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

M1

$\left(2\left(\frac{1}{2} \times \frac{1}{\sqrt{2}}+\frac{\pi}{16}-\frac{1}{8}\right)\right)$

Obtain $\frac{1}{2} \sqrt{2}+\frac{1}{8} \pi-\frac{1}{4}$

A1

ISW
Or exact equivalent from exact working.

Integrate and obtain one of $\frac{1}{2} \ln (2 x+1),-2 \ln (x+2), \frac{-3}{x+2}$

B1 FT

The follow through is on their A, B, C.

Obtain a second term

B1 FT

If the alternative form is used, then either need to use
integration by parts or split the fraction further.

Obtain the third term

B1 FT

Substitute limits correctly in an integral with at least two terms of the form
$\frac{1}{2} \ln (2 x+1),-2 \ln (x+2)$ and $\frac{-3}{x+2}$ and subtract in correct order

M1

The terms used need to have been obtained correctly.
Must be exact values, not decimals.

Obtain $1-\ln 3$

A1