CAIE IGCSE Additional Math 14.12 Integrate Standard Composite Functions Questions

Integrate standard functions with linear inner expressions in CAIE Additional Maths, applying scale factors and using given values to find constants.

Exam points

For a ≠ 0, include the 1/a factor when integrating powers, trigonometric or exponential functions of ax+b.
Use logarithms for 1/(ax+b) terms, including the coefficient scale factor.
Retain an integration constant and use a given point or derivative value to determine it.

[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