NSW Y12 Maths - Advanced Trig Functions Integration of Trig Functions

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Integration of Trig Functions Theory

$\begin{aligned} & \displaystyle \int \cos a x d x=\frac{1}{a} \sin a x+c \\ & \displaystyle \int \sin a x d x=-\frac{1}{a} \cos a x+c \\ & \displaystyle \int \sec ^2 a x d x=\frac{1}{a} \tan a x+c \end{aligned}$\\  \begin{multicols}{2}  \textbf{Example 1}\\%3 (31064) Find \(\displaystyle \int \, \cos\frac{x}{3} \, dx\)\\  \textbf{Example 1 solution}\\ $\begin{aligned} \int \, \cos\frac{x}{3} \, dx &= \frac{1}{\frac{1}{3}} \sin \frac{x}{3} + C\\&= 3\sin \frac{x}{3} +C \end{aligned}$\\  \textbf{Example 2}\\ Find the exact value of \(\displaystyle \int_{-\frac{\pi}{9}}^{\frac{\pi}{9}} \, \sec^2 3x \, dx\)\\  \textbf{Example 2 solution}\\ $\begin{aligned} \displaystyle \int_{-\frac{\pi}{9}}^{\frac{\pi}{9}} \, \sec^2 3x \, dx &= \left[ \frac{1}{3}\tan 3x \right]^{\frac{\pi}{9}}_{-\frac{\pi}{9}}\\&= \frac{1}{3}\left[ \tan\frac{\pi}{3} - \tan\left(\frac{-\pi}{3}\right) \right]\\&= \frac{1}{3} \left[ \tan\frac{\pi}{3} + \tan\frac{\pi}{3} \right] \\ \text{(tan } x\text{ is an odd}& \text{ function, } \tan(-x) = \tan x)\\ &= \frac{2}{3}\tan\frac{\pi}{3}\\&= \frac{2\sqrt{3}}{3} \end{aligned}$\\  \columnbreak  \textbf{Example 3}\\ \begin{itemize} \item[\bf{i)}]Differentiate \(y = {\sin ^2}x\) \item[\bf{ii)}]Hence or otherwise evaluate \(\displaystyle \int\limits_0^{\frac{\pi }{4}} {\sin x\cos x\,\,dx} \) \end{itemize}  \textbf{Example 3 solution}\\ $\begin{aligned} & \begin{aligned}\text { (i) }\quad y &=\sin^{2} x \\ y^{\prime} &=2 \sin x \cos x \end{aligned}\\ &\begin{aligned}\text { (ii) }\quad \int_{0}^{\frac{\pi}{4}} 2 \sin x \cos x d x &=\left[\sin ^{2} x\right]_{0}^{\frac{\pi}{4}} \\ \therefore \int_{0}^{\frac{\pi}{4}} \sin x \cos x d x &=\frac{1}{2}\left[\sin ^{2} x\right]_{0}^{\frac{\pi}{4}} \\ &=\frac{1}{2}\left[\frac{1}{2}-0\right] \\ &=\frac{1}{4} \end{aligned} \end{aligned}$\\  \end{multicols}

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  • Integration of Trig Functions - Video - Integration of trigonometric functions : sin x, cos x and sec² x

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  • Integration of Trig Functions - Video - Definite Integral of Trigonometric Functions

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