### **Integration Cheat Sheet**

#### **1. Basic Antiderivatives:**

- **Power Rule:**
  \[
  \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{(for \(n \neq -1\))}
  \]

- **Constant Rule:**
  \[
  \int a \, dx = ax + C
  \]

- **Exponential Functions:**
  \[
  \int e^x \, dx = e^x + C
  \]
  \[
  \int a^x \, dx = \frac{a^x}{\ln a} + C \quad \text{(for \(a > 0\))}
  \]

- **Logarithmic Functions:**
  \[
  \int \frac{1}{x} \, dx = \ln |x| + C
  \]

#### **2. Trigonometric Integrals:**

- **Basic Trig Functions:**
  \[
  \int \sin(x) \, dx = -\cos(x) + C
  \]
  \[
  \int \cos(x) \, dx = \sin(x) + C
  \]
  \[
  \int \sec^2(x) \, dx = \tan(x) + C
  \]
  \[
  \int \csc^2(x) \, dx = -\cot(x) + C
  \]
  \[
  \int \sec(x) \tan(x) \, dx = \sec(x) + C
  \]
  \[
  \int \csc(x) \cot(x) \, dx = -\csc(x) + C
  \]

- **Inverse Trig Functions:**
  \[
  \int \frac{1}{\sqrt{1-x^2}} \, dx = \arcsin(x) + C
  \]
  \[
  \int \frac{-1}{\sqrt{1-x^2}} \, dx = \arccos(x) + C
  \]
  \[
  \int \frac{1}{1+x^2} \, dx = \arctan(x) + C
  \]

#### **3. Special Integrals:**

- **Integration by Substitution:**
  \[
  \text{If } u = g(x), \text{ then } \int f(g(x))g'(x) \, dx = \int f(u) \, du
  \]

- **Integration by Parts:**
  \[
  \int u \, dv = uv - \int v \, du
  \]
  - Useful for products of functions where one is easier to differentiate and the other to integrate.

- **Trigonometric Substitution:**
  - **For \( \sqrt{a^2 - x^2} \):** Set \(x = a\sin(\theta)\)
  - **For \( \sqrt{a^2 + x^2} \):** Set \(x = a\tan(\theta)\)
  - **For \( \sqrt{x^2 - a^2} \):** Set \(x = a\sec(\theta)\)

- **Partial Fraction Decomposition:**
  - For rational functions, express as a sum of simpler fractions:
  \[
  \frac{P(x)}{Q(x)} = \frac{A}{x-a} + \frac{B}{x-b} + \dots
  \]
  - Integrate each fraction separately.

#### **4. Common Integral Results:**

- **Gaussian Integral:**
  \[
  \int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi}
  \]

- **Integral of \( \sin^2(x) \) and \( \cos^2(x) \):**
  \[
  \int \sin^2(x) \, dx = \frac{x}{2} - \frac{\sin(2x)}{4} + C
  \]
  \[
  \int \cos^2(x) \, dx = \frac{x}{2} + \frac{\sin(2x)}{4} + C
  \]

#### **5. Definite Integrals:**

- **Fundamental Theorem of Calculus:**
  \[
  \int_{a}^{b} f(x) \, dx = F(b) - F(a)
  \]
  - Where \(F(x)\) is the antiderivative of \(f(x)\).

#### **Tips for Integration:**
- **Simplify before integrating**: Factor or expand functions when possible.
- **Recognize standard forms**: Identify patterns from the cheat sheet.
- **Check results by differentiation**: Differentiate your result to confirm it matches the original integrand.

Would you like to explore any of these topics further or need additional examples?

### Related Questions:
1. How do you apply integration by parts in more complex scenarios?
2. What are the key differences between definite and indefinite integrals?
3. How do you handle improper integrals?
4. What is the significance of the Gaussian integral in probability theory?
5. Can you provide more examples of partial fraction decomposition?

### Tip:
When dealing with difficult integrals, try recognizing if the integrand resembles a derivative of a known function or if a substitution can simplify it.

Master integration with our comprehensive cheat sheet. Includes basic antiderivatives, trigonometric integrals, integration techniques like substitution and parts, and common integral results. Ideal for advanced high school and college students.

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