Integrate trigonometric functions including $\sin(kx)$, $\cos(kx)$ and $\sec^2(kx)$, and apply the linear reverse-chain rule for integrals of the form $f(ax+b)$

What this dot point is asking

QCAA wants you to integrate trigonometric functions of the form $\sin(kx)$, $\cos(kx)$ and $\sec^2(kx)$, evaluate definite integrals with exact values at standard angles, and apply trig integration in kinematics and area contexts. The dot point feeds the IA3 PSMT and the EA Paper 1 short response.

Standard trigonometric antiderivatives

The reverse of the Unit 3 trig derivatives, with the $\frac{1}{k}$ reverse-chain factor for non-unit coefficients.

The $\frac{1}{k}$ factor is the reverse of the chain-rule factor that appears when differentiating $\sin(kx)$ to get $k \cos(kx)$. Forgetting this factor is the single most common Paper 1 error in trig integration.

Sign check

The pattern for derivatives is:

• IMATH_20 (no sign change).
• IMATH_21 (sign change).

Reversing:

• IMATH_22 (no sign change).
• IMATH_23 (sign change).

The antiderivative of $\sin$ is $-\cos$. The antiderivative of $\cos$ is $+\sin$. These two patterns generate every trig sign error you'll see.

The linear reverse chain rule

For an integrand of the form $f(ax + b)$ where $a, b$ are constants:

where $F$ is an antiderivative of $f$.

Examples:

• IMATH_32 .
• IMATH_33 .
• IMATH_34 .

The $\frac{1}{a}$ corrects for the chain rule factor that would appear when differentiating the antiderivative.

Definite integrals with exact values

For definite integrals of trig functions evaluated at standard angles, the QCAA Paper 1 expects exact-value answers (in terms of $\pi$, fractions, surds).

Standard exact values:

The Paper 1 expects fluency with these. Substituting $\pi = 3.14159$ and computing decimals loses marks on a Paper 1 exact-value question.

Worked examples

Example 1. Simple definite integral

$\int_{0}^{\pi/4} \sec^2(x) \, dx = [\tan x]_{0}^{\pi/4} = \tan(\pi/4) - \tan(0) = 1 - 0 = 1$.

Example 2. With reverse-chain factor

$\int_{0}^{\pi/6} \cos(3x) \, dx = \left[ \frac{1}{3} \sin(3x) \right]_{0}^{\pi/6} = \frac{1}{3} \sin(\pi/2) - \frac{1}{3} \sin(0) = \frac{1}{3}(1) - 0 = \frac{1}{3}$.

Example 3. Mixed integrand

$\int_{0}^{\pi/2} [2 \sin x + \cos(2x)] \, dx$.

Antiderivative. $-2 \cos x + \frac{1}{2} \sin(2x)$.

Evaluate at $x = \pi/2$. $-2 \cos(\pi/2) + \frac{1}{2} \sin(\pi) = 0 + 0 = 0$.

Evaluate at $x = 0$. $-2 \cos(0) + \frac{1}{2} \sin(0) = -2 + 0 = -2$.

Subtract. $0 - (-2) = 2$.

Area between trig curves

Trig curves often appear in area problems. The general procedure (top minus bottom on the interval between intersection points) is the same as in the area-between-curves dot point.

Example. Area enclosed between $y = \sin x$ and $y = \cos x$ on $[0, \pi/4]$.

At $x = 0$: $\sin 0 = 0$, $\cos 0 = 1$. So $\cos x > \sin x$ on $[0, \pi/4)$.

Area $= \int_{0}^{\pi/4} (\cos x - \sin x) \, dx = [\sin x + \cos x]_{0}^{\pi/4} = (\sin(\pi/4) + \cos(\pi/4)) - (0 + 1) = (\sqrt{2}/2 + \sqrt{2}/2) - 1 = \sqrt{2} - 1$.

Applications in PSMT and EA

The QCAA Methods PSMT often involves modelling periodic phenomena (tides, biological cycles, oscillating systems). Integration of the model gives accumulated quantities (total flow, energy, average values).

The EA Paper 1 short response routinely includes trig integration questions involving the $\frac{1}{k}$ factor.

Common errors

Missing the $\frac{1}{k}$ factor. $\int \sin(2x) \, dx = -\frac{1}{2} \cos(2x) + C$, not $-\cos(2x) + C$. This is the single most common error.

Wrong sign on $\cos$ antiderivative. $\int \sin x = -\cos x$ (sign change); $\int \cos x = +\sin x$ (no sign change).

Using degrees instead of radians. QCAA Methods Paper 1 uses radians. $\sin(\pi/2) = 1$; $\sin(90)$ in calculator mode-degrees is the same value, but the formula expects radians.

Substituting $\pi$ as 3.14 in Paper 1. Paper 1 expects exact values.

Forgetting the constant of integration. Indefinite integrals always include $C$. Definite integrals do not (it cancels in the subtraction).

Sign-cancellation forgotten when integrand changes sign on the interval. For total area, split at zeros and take absolute values; for signed integral, do not.

In one sentence

Integration of trigonometric functions in QCE Methods Unit 4 follows the standard antiderivative rules ($\int \sin(kx) \, dx = -\frac{1}{k} \cos(kx) + C$, $\int \cos(kx) \, dx = \frac{1}{k} \sin(kx) + C$, $\int \sec^2(kx) \, dx = \frac{1}{k} \tan(kx) + C$) with the linear reverse-chain $\frac{1}{a}$ factor; Paper 1 evaluations require exact values at standard angles ($\pi/6, \pi/4, \pi/3, \pi/2, \pi$) and the most-tested detail is including the $\frac{1}{k}$ factor in non-unit coefficient integrals.