Calculus: Essential Guide to Antiderivatives and Integration

📚 Understanding Antiderivatives

In calculus, an antiderivative, also known as an indefinite integral, reverses the process of differentiation. If $F(x)$ is an antiderivative of $f(x)$, then $F'(x) = f(x)$. Finding antiderivatives is a core concept in integration.

📏 Basic Integration Rules

Here are some fundamental integration rules:

• Power Rule: ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C, where n ≠ -1
• Constant Multiple Rule: ∫kf(x) dx = k∫f(x) dx
• Sum/Difference Rule: ∫[f(x) ± g(x)] dx = ∫f(x) dx ± ∫g(x) dx
• Constant Function: ∫k dx = kx + C
• Exponential Function: ∫eˣ dx = eˣ + C

Where 'C' represents the constant of integration.

💡 Techniques of Integration

Several techniques help solve more complex integrals:

1. Substitution (u-substitution): Used when the integral contains a function and its derivative.
2. Integration by Parts: ∫u dv = uv - ∫v du, useful for integrating products of functions.
3. Trigonometric Substitution: Used for integrals involving square roots of the form √(a² - x²), √(a² + x²), or √(x² - a²).
4. Partial Fractions: Used to integrate rational functions by breaking them into simpler fractions.

✍️ Example: Power Rule

Let's find the antiderivative of $f(x) = x^2$:

import sympy as sp

x = sp.Symbol('x')
f = x**2
F = sp.integrate(f, x)

print(F)

Output:

x**3/3

So, ∫x² dx = (x³)/3 + C

➗ Example: U-Substitution

Evaluate ∫2x * cos(x²) dx:

Let u = x², then du = 2x dx

The integral becomes ∫cos(u) du = sin(u) + C

Substitute back: sin(x²) + C

➕ Constant of Integration

Always remember to add the constant of integration, 'C', to the antiderivative, as the derivative of a constant is zero.

⚠️ Important Note

Integration can be more complex than differentiation. Some functions do not have elementary antiderivatives, and numerical methods may be required.