CAIE A-Level Mathematics 3.5.1 Extended Integration Rules

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 $\ln \left|\sec \left(\tan ^{-1} 3\right)\right|$.