CAIE A-Level Mathematics A2 3.8.3 An Initial Condition to Find a Questions

Separate variables: $\int \frac{\sin \theta}{\cos ^{4} \theta} \mathrm{~d} \theta=\int \frac{(x+3)}{\left(x^{2}+9\right)} \mathrm{d} x$
Or $\int \frac{3 \sin \theta}{\cos ^{4} \theta} \mathrm{~d} \theta=\int \frac{3 x+9}{x^{2}+9} \mathrm{~d} x$.

Condone missing integral signs or missing d x or $\mathrm{d} \theta$, but not both.

Obtain $p \sec ^{3} \theta(+A)$
Correct form, p any constant but not 0 .

Use $\int \frac{3 x+9}{x^{2}+9} \mathrm{~d} x=\int\left(\frac{3 x}{x^{2}+9}+\frac{9}{x^{2}+9}\right) \mathrm{d} x$ and obtain $q \ln \left(x^{2}+9\right)$ or
$r \tan ^{-1} \frac{x}{3}(+C)$.

*M1

Might have one third of both sides.

Alt: substitute $x=3 \tan \phi$ to obtain $q \int 1+\tan \phi \mathrm{d} \phi$;
condone if have $\theta$ in place of $\phi$ in this method.

Obtain $q \ln \left(x^{2}+9\right)$ and $r \tan ^{-1} \frac{x}{3}(+C)$
Obtain $q(\phi \mp \ln (\cos \phi))$ OE.

Obtain $\sec ^{3} \theta=\frac{3}{2} \ln \left(x^{2}+9\right)+3 \tan ^{-1} \frac{x}{3}(+C)$ or equivalent
Or might see a third of both sides. Must have 2 different variables.

Use $\theta=\frac{1}{3} \pi, x=3$ in an equation including $p \sec ^{3} \theta, q \ln \left(x^{2}+9\right)$
and $r \tan ^{-1} \frac{x}{3}$ to evaluate the constant of integration
Or as limits in a definite integral.

Limits for $\phi$ are 0 and $\frac{1}{4} \pi$.

Obtain constant $=8-\frac{3}{2} \ln 18-\frac{3}{4} \pi$
OE, e.g. 1.308... to at least 3sf.

Obtain $\cos \theta=0.601$
Marking guidance:

Accept AWRT 0.601.