CAIE IGCSE Additional Math 14.12 Integrate Standard Composite Functions Questions

Practise evaluating integrals, using substitutions and trigonometric identities, and recovering functions from derivative conditions in Cambridge Additional Mathematics problems.

Exam points

Evaluate definite and indefinite integrals, use substitutions and trigonometric forms, and recover functions from derivative conditions.

[Maximum number: 4]

A curve is such that $\frac{\mathrm{d} y}{\mathrm{~d} x}=\sin (3 x+\pi)$ .
The curve passes through the point $\left(\frac{\pi}{9}, \frac{4}{3}\right)$ .
Find the exact y－coordinate of the point on the curve where $x=\frac{5 \pi}{12}$ .

$y: \quad-\frac{1}{3} \cos (3 x+\pi)+c$
B1 for $-\frac{1}{3} \cos (3 x+\pi)$ or for
$k \cos (3 x+\pi)+c$ where k is $\frac{1}{3}$ or k<0

$\frac{4}{3}=-\frac{1}{3} \cos \left(\frac{3 \pi}{9}+\pi\right)+c$
FT their k providing B1 has been awarded

$\left[y=-\frac{1}{3} \cos \left(\frac{5 \pi}{4}+\pi\right)+\frac{7}{6}=\right] \frac{7-\sqrt{2}}{6}$

or exact equivalent

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CAIE IGCSE Additional Math 14.12 Integrate Standard Composite Functions Questions question 1