CAIE A-Level Mathematics 3.5.2 Trigonometric Integration

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.