# Tag Archives: Trigonometric Functions Example 1 Find the derivative of $\sin (2x-5)$ and use this result to deduce $\displaystyle \int 10 \cos (2x-5) dx$. \begin{align} \displaystyle \frac{d}{dx} \sin (2x-5) &= \cos (2x-5) \times \frac{d}{dx} (2x-5) &\color{green}{\text{Chain Rule}} \\ &= \cos (2x-5) \times 2 \\ &= 2 \cos (2x-5) \\ \sin (2x-5) &= \int 2 […] # Trigonometric Integration by Substitution Substitution of Angle Parts Example 1 Find \displaystyle \int{(2x+3) \sin (x^2+3x)}dx. \( \begin{align} \displaystyle \text{Let } u &= x^2+3x \\ \dfrac{du}{dx} &= 2x + 3 \\ du &= (2x+3)dx \\ \int{(2x+3) \sin (x^2+3x)}dx &= \int{\sin u}du \\ &= -\cos u + C \\ &= -\cos {(x^2+3x)} + C \\ \end{align} Substitution of $\sin{x}$ $$[…] # Integration of Trigonometric Functions Integration of \cos{x}$$ \begin{align} \displaystyle \dfrac{d}{dx}\sin{x} &= \cos{x} \\ \sin{x} &= \int{\cos{x}}dx \\ \therefore \int{\cos{x}}dx &= \sin{x} +c \\ \end{align} $$Integration of \cos{(ax+b)}$$ \begin{align} \displaystyle \dfrac{d}{dx}\sin{(ax+b)} &= \cos{(ax+b)} \times \dfrac{d}{dx}(ax+b) \\ &= \cos{(ax+b)} \times a \\ &= a\cos{(ax+b)} \\ \sin{(ax+b)} &= \int{a\cos{(ax+b)}}dx \\ &= a\int{\cos{(ax+b)}}dx \\ \dfrac{1}{a}\sin{(ax+b)} &= \int{\cos{(ax+b)}}dx \\ \therefore \int{\cos{(ax+b)}}dx […] # Maxima and Minima with Trigonometric Functions

Periodic motions can be modelled by a trigonometric equation. By differentiating these functions we are then able to solve problems relating to maxima (maximums) and minima (minimums). Remember that the following steps are used when solving a maximum or minimum problem. Step 1: Find $f^{\prime}(x)$ to obtain the gratest function. Step 2: Solve for $x$ […] 