Antidifferentiation and Basic Integration Rules

Antidifferentiation

Basic Integration Rules

The basic rules for integrals of functions are as follows:

1. Constant Rule
      - Differentiation Formula:
   $\frac{d}{dx}[C] = 0$
      - Integration Formula:
   $\int 0 \, dx = C$

2. Constant Coefficient Rule
      - Differentiation Formula:
   $\frac{d}{dx}[k x] = k$
      - Integration Formula:
   $\int k \, dx = kx + C$

3. Power Rule (for non-zero n)
      - Differentiation Formula:
   $\frac{d}{dx}[x^n] = nx^{n-1}$
      - Integration Formula:
   $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \text{ if } n \neq -1$
      - Special Case (n = -1):
   $\int x^{-1} \, dx = \int \frac{1}{x} \, dx = \ln x + C$

4. Constant Multiple Rule
      - Differentiation Formula:
   $\frac{d}{dx}[k f(x)] = k f'(x)$
      - Integration Formula:
   $\int k f(x) \, dx = k \int f(x) \, dx$

5. Sum and Difference Rule
      - Differentiation Formula:
   $\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$
      - Integration Formula:
   $\int [f(x) \pm g(x)] \, dx = \int f(x) \, dx \pm \int g(x) \, dx$

6. Exponential Function Rule
      - Differentiation Formula:
   $\frac{d}{dx}[e^x] = e^x$
      - Integration Formula:
   $\int e^x \, dx = e^x + C$

Examples of Antidifferentiation

1. Example:
   $\int \sin(x^2) \, dx$

2. Example:
   $\int (\sin(\pi x) - \sin(3x)) \, dx$

3. Example:
   $\int \cos(4x) \, dx$

4. Example:
   $\int \tan(2x) \sec(2x) \, dx$

5. Rewriting before Integrating:
   $\int \sqrt{x^3} (x - 4) \, dx$

6. Example:
   $\int \left( \frac{\sin x}{1 - \sin^2 x} \right) \, dx$

7. Example:
   $\int (x+1) \sqrt{x} \, dx$

8. Example:
   $\int (2 \cos^2 x - 1) \, dx$