Cards In This Deck

• $$\begin{gathered} \text{Under what condition is the integration rule }\int x^n \\ ,dx=\frac{x^{n+1}}{n+1}\text{ **invalid**?} \end{gathered}$$
  $$\begin{gathered} \text{When **}n=-1\text{**} \\ \text{(i.e., simplifying to }\frac{1}{x}\text{).} \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \int \frac{3}{x^2} \\ ,dx=\frac{3x^{-3}}{-3}+c \end{gathered}$$
  $$\begin{gathered} \text{❌ **False**} \\ \text{✅ **Correct Answer:**} \\ \text{First write as }3x^{-2}\text{.} \\ \text{Add 1 to power: }-2+1=-1\text{.} \\ \text{Divide by new power: }\frac{3x^{-1}}{-1}=-3x^{-1}+c\text{ (or }-\frac{3}{x}+c\text{).} \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \int 5 \\ ,dx=0 \end{gathered}$$
  $$\begin{gathered} \text{❌ **False**} \\ \text{✅ **Correct Answer:** }5x+c \\ \text{The derivative of a constant is 0, but the **integral** of a constant }k\text{ is }kx+c\text{.} \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \int 2x^3 \\ ,dx=\frac{2x^4}{2}+c \end{gathered}$$
  $$\begin{gathered} \text{❌ **False**} \\ \text{✅ **Correct Answer:** }\frac{2x^4}{4}+c=\frac{1}{2}{x}^4+c \\ \text{You must divide by the **new power** (}3+1=4\text{), not the coefficient (}2\text{).} \end{gathered}$$
• $$\begin{gathered} \text{What is the general formula for integrating a power of }x\text{?} \\ \int x^n \\ ,dx=\dots \end{gathered}$$
  $\frac{x^{n+1}}{n+1}+c$
• $$\begin{gathered} \text{What is the **first step** to integrate the following expression?} \\ \int \frac{(x+2{)}^2}{\sqrt{x}} \\ ,dx \end{gathered}$$
  $$\begin{gathered} \text{**Simplify the integrand** into a sum of terms in the form }a{x}^n\text{.} \\ \text{1. **Expand** the numerator: }{x}^2+4x+4 \\ \text{2. **Split** the fraction: }\frac{x^2}{x^{0.5}}+\frac{4x}{x^{0.5}}+\frac{4}{x^{0.5}} \\ \text{3. **Simplify** indices: }{x}^{1.5}+4x^{0.5}+4x^{-0.5} \end{gathered}$$
• $$\begin{gathered} \text{How do you write }\int \frac{1}{2\sqrt{x}} \\ ,dx\text{ in a form suitable for integration?} \end{gathered}$$
  $$\begin{gathered} \text{Convert the root to a fractional power and move it to the numerator:} \\ y=\frac{1}{2}{x}^{-\frac{1}{2}} \\ \text{Then apply the power rule.} \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \text{To integrate }\int (3x-1{)}^2 \\ ,dx\text{, you should use the reverse chain rule to get }\frac{(3x-1{)}^3}{3}+c\text{?} \end{gathered}$$
  $$\begin{gathered} \text{❌ **False** (It leads to the wrong coefficient)} \\ \text{✅ **P1 Method:**} \\ \text{**Expand** the brackets first: }\int (9x^2-6x+1) \\ ,dx \\ \text{Then integrate term by term: }3x^3-3x^2+x+c \end{gathered}$$