Test Questions on Integrating Polynomials Using the Power Rule

📚 Quick Study Guide

• 🔢 The Power Rule: The power rule for integration states that $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, where $n \neq -1$ and $C$ is the constant of integration.
• ➕ Constant Multiple Rule: The integral of a constant times a function is the constant times the integral of the function: $\int kf(x) dx = k \int f(x) dx$.
• ✨ Sum/Difference Rule: The integral of a sum or difference of functions is the sum or difference of their integrals: $\int [f(x) \pm g(x)] dx = \int f(x) dx \pm \int g(x) dx$.
• 🛑 Constant of Integration: Always remember to add the constant of integration, $C$, to the result of indefinite integrals.

Practice Quiz

1. What is the integral of $x^3$ with respect to $x$?

   1. $\frac{x^4}{4} + C$
   2. $3x^2 + C$
   3. $\frac{x^2}{2} + C$
   4. $x^4 + C$

2. What is the integral of $5x^2$ with respect to $x$?

   1. $\frac{5x^3}{3} + C$
   2. $10x + C$
   3. $5x^3 + C$
   4. $\frac{x^3}{3} + C$

3. What is the integral of $2x + 3$ with respect to $x$?

   1. $x^2 + 3x + C$
   2. $2 + C$
   3. $2x^2 + 3x + C$
   4. $x^2 + C$

4. What is the integral of $x^{-2}$ with respect to $x$?

   1. $-\frac{1}{x} + C$
   2. $\frac{1}{x} + C$
   3. $-2x^{-3} + C$
   4. $x^{-1} + C$

5. What is the integral of $4x^3 - 2x + 1$ with respect to $x$?

   1. $x^4 - x^2 + x + C$
   2. $12x^2 - 2 + C$
   3. $4x^4 - 2x^2 + x + C$
   4. $x^4 - x^2 + C$

6. What is the integral of $\frac{1}{2}x^4$ with respect to $x$?

   1. $\frac{x^5}{10} + C$
   2. $2x^3 + C$
   3. $\frac{x^5}{2} + C$
   4. $\frac{5x^3}{2} + C$

7. What is the integral of $7x^6$ with respect to $x$?

   1. $x^7 + C$
   2. $42x^5 + C$
   3. $\frac{7x^7}{7} + C$
   4. $7x^7 + C$