Cards In This Deck

• $$\begin{gathered} \text{What is the result of the integral:} \\ \int e^{ax+b} \\ ,dx \end{gathered}$$
  $\frac{1}{a}{e}^{ax+b}+c$
• $$\begin{gathered} \text{In a modelling question, you derive the differential equation }\frac{dP}{dt}=kP\text{.} \\ \text{What is the general form of the solution for }P\text{ in terms of }t\text{?} \end{gathered}$$
  $P=A{e}^{kt}$
• $$\begin{gathered} \text{True or False?} \\ \int \frac{1}{(3x+2{)}^3} \\ ,dx=\ln |3x+2{|}^3+c \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Method: Write as powers and use the reverse chain rule:} \\ \int (3x+2{)}^{-3} \\ ,dx=\frac{(3x+2{)}^{-2}}{-2\times 3}+c \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \int \frac{1}{x} \\ ,dx=\ln (x)+c \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Answer:} \\ \ln |x|+c \\ \text{The modulus signs are essential.} \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ {\int }_0^{ \\ \pi /4}\cos 2x \\ ,dx\text{ can be evaluated using limits in degrees.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{When integrating or differentiating trigonometric functions, the limits must be in radians.} \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \int \cos (x^2+1) \\ ,dx=\frac{1}{2x}\sin (x^2+1)+c \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{This cannot be integrated using the standard linear rule because the inner function (}{x}^2+1\text{) is non-linear.} \end{gathered}$$
• $$\begin{gathered} \text{Which function should you consider differentiating to solve:} \\ \int \cot x \\ ,{cosec}^{2}x \\ ,dx \end{gathered}$$
  $$\begin{gathered} \text{Consider }y={\cot }^{2}x \\ \text{OR} \\ \text{Consider }y={cosec}^{2}x \end{gathered}$$
• $$\begin{gathered} \text{What is the standard procedure to integrate a function in the form:} \\ \int k{f}^{\prime }(x)[f(x){]}^n \\ ,dx \end{gathered}$$
  $$\begin{gathered} \text{1. Consider the function }y=[f(x){]}^{n+1}\text{.} \\ \text{2. Differentiate }y\text{ to check the coefficient.} \\ \text{3. Adjust the constant }k\text{ to match the integral.} \end{gathered}$$
• $$\begin{gathered} \text{Which integration method should be used for these two integrals?} \\ \text{1. }\int \frac{x}{x^2+4} \\ ,dx \\ \text{2. }\int \frac{1}{x^2+4} \\ ,dx \end{gathered}$$
  $$\begin{gathered} \text{1. Reverse Chain Rule / Logarithms: The numerator (}x\text{) is related to the derivative of the denominator (}2x\text{).} \\ \text{2. Not Logarithms: The numerator (}1\text{) does not contain the derivative of }{x}^2\text{. (Note: In P3, you usually cannot integrate this yet, or it simplifies via P4 standard forms/substitution).} \end{gathered}$$
• $$\begin{gathered} \text{Which trigonometric identity should you use to integrate this function?} \\ \int {\cos }^{2}x \\ ,dx \end{gathered}$$
  $$\begin{gathered} \text{Double Angle Formula:} \\ \cos 2x=2{\cos }^{2}x-1 \\ \text{Rearrange to:} \\ {\cos }^{2}x=\frac{1}{2}(1+\cos 2x) \end{gathered}$$
• $$\begin{gathered} \text{Which trigonometric identity should you use to integrate this function?} \\ \int {\tan }^{2}x \\ ,dx \end{gathered}$$
  $$\begin{gathered} {\sec }^{2}x=1+{\tan }^{2}x \\ \text{Rearrange to:} \\ {\tan }^{2}x={\sec }^{2}x-1 \end{gathered}$$
• $$\begin{gathered} \text{To integrate }\int \sin 3x\cos 3x \\ ,dx\text{, name two valid methods.} \end{gathered}$$
  $$\begin{gathered} \text{1. Double Angle Identity: Rewrite as }\frac{1}{2}\sin 6x\text{.} \\ \text{2. Reverse Chain Rule: Treat as }\sin 3x\times (\sin 3x{)}^{\prime }\text{ to get }\frac{1}{6}{\sin }^{2}3x\text{.} \end{gathered}$$
• $$\begin{gathered} \text{What is the best way to start integrating:} \\ \int (\sec x+\tan x{)}^2 \\ ,dx \end{gathered}$$
  $$\begin{gathered} \text{Expand the brackets first.} \\ \text{Result: }{\sec }^{2}x+2\sec x\tan x+{\tan }^{2}x \\ \text{Then use the identity }{\tan }^{2}x={\sec }^{2}x-1\text{.} \end{gathered}$$