Cards In This Deck

• $$\begin{gathered} \text{True or False?} \\ \int 3x^2 \\ ,dx=x^3 \end{gathered}$$
  $$\begin{gathered} \text{❌ **False**} \\ \text{✅ **Correct Answer:** }{x}^3+c \\ \text{Indefinite integrals represent a family of curves. You must **always** add the constant of integration, }c\text{.} \end{gathered}$$
• $$\begin{gathered} \text{You are given the gradient function }{f}^{\prime }(x)\text{ and a point }P(3,2)\text{ that the curve passes through.} \\ \text{What is the specific purpose of the point }P\text{ in this problem?} \end{gathered}$$
  $$\begin{gathered} \text{It is used to **find the value of the constant of integration (}c\text{)**.} \\ \text{1. Integrate }{f}^{\prime }(x)\text{ to get }f(x)+c\text{.} \\ \text{2. Substitute }x=3\text{ and }f(x)=2\text{.} \\ \text{3. Solve for }c\text{.} \end{gathered}$$