Sure, here are 10 multiple-choice questions related to integration:
1. **What is the integral of \( \int 3x^2 \, dx \)?**
- A) \( x^3 + C \)
- B) \( x^3 + 3x + C \)
- C) \( x^3 + 3x^2 + C \)
- D) \( x^3 + C \)
2. **Evaluate \( \int e^x \, dx \).**
- A) \( e^x + C \)
- B) \( e^x \)
- C) \( \frac{e^x}{x} + C \)
- D) \( e^{x+1} + C \)
3. **Find the integral \( \int \frac{1}{x} \, dx \).**
- A) \( \ln|x| + C \)
- B) \( \ln(x) + C \)
- C) \( \frac{1}{x} + C \)
- D) \( x + C \)
4. **What is the integral of \( \int \sin(x) \, dx \)?**
- A) \( -\cos(x) + C \)
- B) \( \cos(x) + C \)
- C) \( -\sin(x) + C \)
- D) \( \sin(x) + C \)
5. **Evaluate \( \int x e^x \, dx \) using integration by parts.**
- A) \( x e^x - e^x + C \)
- B) \( x e^x + e^x + C \)
- C) \( e^x (x - 1) + C \)
- D) \( e^x (x + 1) + C \)
6. **What is the integral \( \int \cos^2(x) \, dx \)?**
- A) \( \frac{x}{2} + \frac{\sin(2x)}{4} + C \)
- B) \( \frac{x}{2} - \frac{\sin(2x)}{4} + C \)
- C) \( \frac{x}{2} + \sin(x) \cos(x) + C \)
- D) \( \frac{x}{2} - \sin(x) \cos(x) + C \)
7. **Find the integral \( \int \frac{1}{1+x^2} \, dx \).**
- A) \( \tan^{-1}(x) + C \)
- B) \( \ln|1+x^2| + C \)
- C) \( \frac{1}{x} + C \)
- D) \( \arctan(x) + C \)
8. **Evaluate \( \int \frac{2x}{1+x^2} \, dx \).**
- A) \( \ln|1+x^2| + C \)
- B) \( \ln|x| + C \)
- C) \( \arctan(x) + C \)
- D) \( \frac{1}{2} \ln|1+x^2| + C \)
9. **What is the integral \( \int \sec^2(x) \, dx \)?**
- A) \( \sec(x) + C \)
- B) \( \ln|\sec(x) + \tan(x)| + C \)
- C) \( \tan(x) + C \)
- D) \( \sec(x) + \tan(x) + C \)
10. **Find \( \int x^2 \ln(x) \, dx \) using integration by parts.**
- A) \( \frac{x^3 \ln(x)}{3} - \frac{x^3}{9} + C \)
- B) \( \frac{x^3 \ln(x)}{3} - \frac{x^3}{6} + C \)
- C) \( \frac{x^3 \ln(x)}{3} - \frac{x^3 \ln(2)}{6} + C \)
- D) \( \frac{x^3 \ln(x)}{3} - \frac{x^3}{6} + \frac{x^3}{9} + C \)
Feel free to use or modify these questions as needed!
