Cards In This Deck

• $$\begin{gathered} \text{Procedure to integrate }\int x\sqrt{x-2} \\ ,dx\text{ using the substitution }u=x-2\text{?} \end{gathered}$$
  $$\begin{gathered} \text{1. Replace }dx\text{ with }du\text{.} \\ \text{2. Replace }\sqrt{x-2}\text{ with }{u}^{1/2}\text{.} \\ \text{3. Rearrange }u=x-2\text{ to }x=u+2\text{ to replace the remaining }x\text{.} \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \text{If limits for a substitution integral turn out to be }u=5\text{ (lower) and }u=2\text{ (upper), you must swap them to make the smaller number the lower limit.} \end{gathered}$$
  $\text{❌ False}$
• $$\begin{gathered} \text{True or False?} \\ \text{When using integration by substitution }u=2x+1\text{ for the definite integral }{\int }_0^{4}f(x) \\ ,dx\text{, the limits remain }0\text{ and }4\text{.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{You must convert the }x\text{-limits to }u\text{-limits.} \end{gathered}$$
• $$\begin{gathered} \text{To integrate }\int x{e}^{x^2} \\ ,dx\text{, would you use Integration by Parts or Substitution?} \end{gathered}$$
  $\text{Substitution (}u=x^2\text{)}$
• $$\begin{gathered} \text{When using Integration by Parts for }\int x\ln x \\ ,dx\text{, which term should you choose as }u\text{?} \end{gathered}$$
  $u=\ln x$
• $$\begin{gathered} \text{When using Integration by Parts for }\int x{e}^x \\ ,dx\text{, which term should you choose as }u\text{?} \end{gathered}$$
  $u=x$
• $$\begin{gathered} \text{What is the standard method to integrate }\int \ln x \\ ,dx\text{?} \end{gathered}$$
  $\text{Use Integration by Parts with }u=\ln x\text{ and }\frac{dv}{dx}=1\text{.}$
• $$\begin{gathered} \text{What strategy is needed to integrate }\int e^x\sin x \\ ,dx\text{?} \end{gathered}$$
  $\text{Use Integration by Parts twice.}$
• $$\begin{gathered} \text{True or False?} \\ \text{The formula for Integration by Parts is:} \\ \int u\frac{dv}{dx}dx=uv+\int v\frac{du}{dx}dx \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{Correct Formula: }uv-\int v\frac{du}{dx}dx \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \int \frac{1}{(2x+1{)}^3} \\ ,dx=\ln |(2x+1{)}^3|+C \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{Correct method: Use the Power Rule.} \\ \int (2x+1{)}^{-3} \\ ,dx=\frac{(2x+1{)}^{-2}}{-2\times 2}+C \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \int \frac{1}{x} \\ ,dx=\ln x+C \end{gathered}$$
  $$\begin{gathered} \text{❌ False (Technically incomplete)} \\ \text{Correct Answer: }\ln |x|+C \end{gathered}$$
• $\text{Are you required to calculate volumes of revolution formed by rotating a curve around the }y\text{-axis in Unit P4?}$
  $\text{No.}$
• $\text{What formula is used to find the Volume of Revolution when a curve defined parametrically by }x=f(t),y=g(t)\text{ is rotated around the }x\text{-axis?}$
  $$\begin{gathered} V= \\ \pi {\int }_{t_1}^{t_2}{y}^2\frac{dx}{dt} \\ ,dt \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \text{The area under a curve defined by parametric equations }x=f(t),y=g(t)\text{ is given by:} \\ \int y \\ ,dt \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{The correct formula is:} \\ \int y\frac{dx}{dt} \\ ,dt \end{gathered}$$
• $$\begin{gathered} \text{How do you handle the limits when calculating the Area under a parametric curve }{\int }_{x=a}^{x=b}y \\ ,dx\text{?} \end{gathered}$$
  $\text{Solve }x(t)=a\text{ and }x(t)=b\text{ to find the corresponding values of }{t}_1\text{ and }{t}_2\text{.}$
• $\text{How do you find the Particular Solution of a differential equation?}$
  $$\begin{gathered} \text{1. Find the General Solution (with constant }C\text{).} \\ \text{2. Substitute the given boundary conditions (e.g., }y=2\text{ when }x=0\text{).} \\ \text{3. Solve for }C\text{ and rewrite the final equation.} \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \text{If a differential equation solution is }\ln |y|=x+C\text{, then }y=e^x+e^C\text{.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{Correct Algebra: }y=e^{x+C}=e^x\cdot e^C=A{e}^x \end{gathered}$$
• $\text{First step to solve the differential equation }\frac{dy}{dx}=xy+y\text{?}$
  $\text{Factorise the RHS to }y(x+1)\text{.}$
• $$\begin{gathered} \text{What is the correct form for the partial fraction decomposition of:} \\ \frac{5x+1}{(x+1)(x-2{)}^2}\text{?} \end{gathered}$$
  $\frac{A}{x+1}+\frac{B}{x-2}+\frac{C}{(x-2{)}^2}$
• $$\begin{gathered} \text{First step to integrate a rational function where the degree of the numerator is }\ge \text{ the degree of the denominator?} \\ \text{Example: }\int \frac{x^2}{x-1} \\ ,dx \end{gathered}$$
  $\text{Use Algebraic Division (Long Division) first.}$