Step-by-Step Indefinite Integral Examples

📚 Quick Study Guide

• 🔍 An indefinite integral represents the antiderivative of a function. It includes an arbitrary constant of integration, denoted as $C$.
• 💡 The power rule for integration states that $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, where $n \neq -1$.
• 📝 The integral of a constant $k$ is given by $\int k dx = kx + C$.
• ➕ The integral of a sum or difference is the sum or difference of the integrals: $\int [f(x) \pm g(x)] dx = \int f(x) dx \pm \int g(x) dx$.
• 🌱 Remember to always add the constant of integration $C$ to the result of an indefinite integral.
• 🔄 Integration is the reverse process of differentiation.
• 📈 The indefinite integral represents a family of functions, all differing by a constant.

🧪 Practice Quiz

1. Question 1: What is the indefinite integral of $x^2$?
   1. $\frac{x^3}{2} + C$
   2. $\frac{x^3}{3} + C$
   3. $2x + C$
   4. $x^3 + C$
2. Question 2: Find the indefinite integral of $5$ with respect to $x$.
   1. $5x^2 + C$
   2. $5 + C$
   3. $5x + C$
   4. $C$
3. Question 3: Evaluate $\int (3x^2 + 2x) dx$.
   1. $x^3 + x^2 + C$
   2. $6x + 2 + C$
   3. $x^3 + x^2 + x + C$
   4. $3x^3 + 2x^2 + C$
4. Question 4: What is the indefinite integral of $\frac{1}{x}$?
   1. $x^{-2} + C$
   2. $\ln|x| + C$
   3. $-\frac{1}{x^2} + C$
   4. $\frac{1}{x^2} + C$
5. Question 5: Determine $\int (4x^3 - 6x^2 + 2x - 3) dx$.
   1. $x^4 - 2x^3 + x^2 - 3x + C$
   2. $12x^2 - 12x + 2 + C$
   3. $4x^4 - 6x^3 + 2x^2 - 3x + C$
   4. $x^3 - x^2 + x - 3 + C$
6. Question 6: What is the indefinite integral of $e^x$?
   1. $\frac{e^{x+1}}{x+1} + C$
   2. $e^x + C$
   3. $xe^x + C$
   4. $\ln(x) + C$
7. Question 7: Evaluate $\int (2\sin(x) + 3\cos(x)) dx$.
   1. $-2\cos(x) + 3\sin(x) + C$
   2. $2\cos(x) - 3\sin(x) + C$
   3. $-2\sin(x) + 3\cos(x) + C$
   4. $2\sin(x) - 3\cos(x) + C$