November 18, 2024
\[\begin{align*} \boxed{\int \sec(2x) \: dx = \frac{1}{2}\ln \left|\tan \left(2x\right)+\sec \left(2x\right)\right|+C} \end{align*}\] where \( C \) is the constant of integration. Solving for the Integral To solve many integrals that involve composite functions, we turn to $u$-substitution. It is a little easier to work with $\sec(x)$ instead so we set $u = 2x$ and $du = […]
November 15, 2024
Introduction to Radian and Degree Conversion Trigonometry often uses radians as a way to measure angles, although degrees are also common in many applications. Knowing how to convert between these units is essential. The main relationship between radians and degrees is: \[\begin{align*} \pi \text{ radians} = 180^\circ \end{align*}\] If we divide both sides by 3 […]
November 15, 2024
Cosine of $\pi$ is $\boxed{-1}$. 1 Unit Circle There are multiple ways to find $\cos \pi$. One option is to use the unit circle: Notice that $\cos \pi = \cos 180^\circ$. Recall that $180^\circ$ is a straight angle, so the terminal side in standard position would intersect the unit circle on the $x$-axis at $(-1,0)$. […]
November 15, 2024
Unit Circle The unit circle is defined by $x^2 + y^2 = 1$, which is a circle with radius $1$, centered at the origin $(0, 0)$. An example of the unit circle is below: Standard Position of an Angle An angle is in standard position if its initial side is on the positive $x$-axis, its […]
November 15, 2024
The unit circle is defined as $x^2 + y^2 = 1$, which is a circle with radius $1$, centered at the origin $(0, 0)$. It is used in trigonometry to simplify finding values of trig functions. An empty unit circle is below with markings to be filled in: Blank_Unit_Circle.PDF Practice Problems Find the radian value […]
November 8, 2024
\[\begin{align*} \boxed{\int 2x \ln(x) \, dx = x^2 \ln(x) – \frac{x^2}{2} + C} \end{align*}\] where \( C \) is the constant of integration. Introduction In calculus, integrating functions that combine both polynomial and logarithmic terms, like \( 2x \ln(x) \), requires the specific technique of integration by parts. This method is particularly useful for integrating […]
November 8, 2024
\[\begin{align*} \boxed{\frac{d}{dx} \left( 2x \ln(x) \right) = 2 \ln(x) + 2} \end{align*}\] Introduction To find the derivative of \( 2x \ln(x) \), we will use the product rule. Learning to work with logarithmic expressions is important in calculus as they show up later when tackling integration. Step-by-Step Solution The product rule, which states that if […]
November 8, 2024
\[\begin{align*} \boxed{6^3 = 6 \times 6 \times 6 = 216} \end{align*}\] Introduction The expression \( 6^3 \) is an example of exponential notation, where: – \( 6 \) is the base, which is the number being multiplied. – \( 3 \) is the exponent, which tells us how many times to multiply the base by […]
November 8, 2024
\[\begin{align*} \boxed{10^5 = 10 \times 10 \times 10 \times 10 \times 10 = 100,000} \end{align*}\] Introduction Exponents provide a compact way to express large numbers through repeated multiplication. The expression \( 10^5 \) is an example of exponential notation, where: – \( 10 \) is the base, representing the number being multiplied. – \( 5 […]
