Advanced U-Substitution Quiz for Exponential and Logarithmic Integrals

📚 Quick Study Guide

• Ⅻ U-Substitution is a technique to simplify integrals by substituting a function with a new variable, $u$.
• 💡 For exponential functions like $e^{f(x)}$, consider $u = f(x)$, then $du = f'(x) dx$.
• 📝 For logarithmic functions like $\ln(f(x))$, consider $u = \ln(f(x))$, then $du = \frac{f'(x)}{f(x)} dx$.
• ➕ Remember the integral of $e^u$ is $e^u + C$, and the integral of $\frac{1}{u}$ is $\ln|u| + C$.
• 🔑 Don't forget to change the limits of integration if you're working with definite integrals!

🧪 Practice Quiz

1. What should $u$ be if you are solving $\int x e^{x^2} dx$ using u-substitution?
   1. $u = e^{x^2}$
   2. $u = x$
   3. $u = x^2$
   4. $u = e^x$
2. Evaluate $\int 2x e^{x^2} dx$.
   1. $e^{x^2} + C$
   2. $x^2 e^{x^2} + C$
   3. $e^x + C$
   4. $2 e^{x^2} + C$
3. Solve the integral $\int \frac{1}{x \ln(x)} dx$.
   1. $\ln|\ln(x)| + C$
   2. $\frac{1}{\ln(x)} + C$
   3. $\ln|x| + C$
   4. $\frac{\ln(x)}{x} + C$
4. What is the proper substitution for $\int \frac{x}{x^2 + 1} dx$?
   1. $u = x$
   2. $u = x^2 + 1$
   3. $u = \frac{1}{x^2 + 1}$
   4. $u = x^2$
5. Evaluate $\int \frac{x}{x^2 + 1} dx$.
   1. $\frac{1}{2} \ln|x^2 + 1| + C$
   2. $\ln|x^2 + 1| + C$
   3. $\frac{1}{x^2 + 1} + C$
   4. $\frac{x^2}{2} + C$
6. Solve the definite integral $\int_{1}^{e} \frac{(\ln x)^2}{x} dx$.
   1. $\frac{1}{3}$
   2. $1$
   3. $\frac{2}{3}$
   4. $0$
7. Evaluate $\int e^x \sqrt{1 + e^x} dx$.
   1. $\frac{2}{3} (1 + e^x)^{3/2} + C$
   2. $2 (1 + e^x)^{3/2} + C$
   3. $\frac{1}{2\sqrt{1 + e^x}} + C$
   4. $\sqrt{1 + e^x} + C$