Tricks for remembering integral rules

by Kurious Fox

Understanding the relationship between derivatives and antiderivatives can significantly help in remembering and applying the rules for finding antiderivatives (also known as integrals). Here’s how this relationship aids in comprehension and recall:

1. Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links derivatives and integrals, stating that:

  1. Part 1: If F is an antiderivative of f on an interval I , then:
    \int_a^b f(x) \, dx = F(b) - F(a)
  2. Part 2: If f is continuous on [a, b] and F is defined by:
    F(x) = \int_a^x f(t) \, dt
    then F is differentiable on (a, b) and F'(x) = f(x) .

Understanding this theorem helps solidify that differentiation and integration are inverse processes. This can guide your understanding and recall of rules, as performing integration essentially involves reversing differentiation.

2. Common Derivative-Antiderivative Pairs

Knowing common derivative-antiderivative pairs can help you remember the corresponding rules. For example:

  • If \frac{d}{dx}(x^n) = nx^{n-1} , then \int x^{n-1} \, dx = \frac{x^n}{n} + C for n \neq 0 .
  • If \frac{d}{dx}(\sin x) = \cos x , then \int \cos x \, dx = \sin x + C .
  • If \frac{d}{dx}(\cos x) = -\sin x , then \int -\sin x \, dx = \cos x + C .

3. Rules for Differentiation and Their Inverses

Many rules for differentiation have corresponding rules for integration:

  • Power Rule:
  • Derivative: \frac{d}{dx} x^n = nx^{n-1}
  • Antiderivative: \int x^n \, dx = \frac{x^{n+1}}{n+1} + C (for n \neq -1 )
  • Sum Rule:
  • Derivative: \frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x)
  • Antiderivative: \int [f(x) + g(x)] \, dx = \int f(x) \, dx + \int g(x) \, dx
  • Constant Multiple Rule:
  • Derivative: \frac{d}{dx} [cf(x)] = c f'(x)
  • Antiderivative: \int cf(x) \, dx = c \int f(x) \, dx

4. Integration by Substitution (Reverse of Chain Rule)

The chain rule states that \frac{d}{dx} f(g(x)) = f'(g(x))g'(x) . When integrating, we often use substitution, which is essentially the reverse process:

  • If u = g(x) , then du = g'(x) dx .
  • This allows us to rewrite \int f(g(x))g'(x) \, dx as \int f(u) \, du .

5. Integration by Parts (Reverse of Product Rule)

The product rule states that \frac{d}{dx} [u(x)v(x)] = u'(x)v(x) + u(x)v'(x) . Integration by parts is the reverse process:

  • If u = f(x) and dv = g(x) \, dx , then:
    \int u \, dv = uv - \int v \, du

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