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Basic Antidifferentiation formulas

Definition

Basic antidifferentiation formulas are standardized rules derived by reversing well-known differentiation patterns. They provide a “lookup table” for converting common rates of change back into their original functions.

Why It Matters

They are the lookup table that makes calculus a practical tool for daily engineering and physics. Without these pre-derived patterns, every integration would be a manual struggle, making modern technological design impossibly slow.

Core Concepts

  • Power Rule (reverse of the differentiation power rule) ∫xn dx=xn+1n+1+C(n≠−1)\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)

    • How to read: “The integral of x to the n, with respect to x, equals x to the power of the quantity n plus one, all over n plus one, plus C.”
    • Meaning: Increase the exponent by 1, divide by the new exponent. The +C appears because the derivative of any constant is zero — every antiderivative is actually a whole family of functions differing by a constant. This is the workhorse rule for polynomials. Remember: the power rule for integration fails at n = −1 (you would divide by zero).

  • Natural Logarithm — the missing power rule case ∫1x dx=ln⁡∣x∣+C\int \frac{1}{x} \, dx = \ln |x| + C

    • How to read: “The integral of one over x, with respect to x, equals the natural logarithm of the absolute value of x, plus C.”
    • Meaning: This is the exact case the power rule cannot handle. It is also the definition of the natural logarithm in calculus. Always use the absolute value because the domain of ln is positive, but 1/x is defined for negative x as well (the antiderivative on negative reals is ln(−x) + C).

  • Exponential ∫ekx dx=1kekx+C(k≠0)\int e^{kx} \, dx = \frac{1}{k} e^{kx} + C \quad (k \neq 0)

    • How to read: “The integral of e to the k x, with respect to x, equals one over k times e to the k x, plus C.”
    • Meaning: The exponential is its own derivative (scaled by k). The 1/k factor is the chain-rule reversal. Ubiquitous in growth, decay, compound interest, RC circuits, etc.

  • Sine and Cosine (and their multiples) ∫sin⁡kx dx=−1kcos⁡kx+C,∫cos⁡kx dx=1ksin⁡kx+C\int \sin kx \, dx = -\frac{1}{k}\cos kx + C, \quad \int \cos kx \, dx = \frac{1}{k}\sin kx + C

    • How to read: “The integral of the sine of k x, with respect to x, equals negative one over k times the cosine of k x, plus C”; “the integral of the cosine of k x, with respect to x, equals one over k times the sine of k x, plus C.”
    • Meaning: Direct reversal of the derivatives you already know. The k in the denominator comes from the chain rule. For plain sin x or cos x (k=1) the factor is just ±1. These plus the power-reduction identities (from double-angle) let you integrate almost any integer power of sin or cos.

  • General advice on +C: Always add the arbitrary constant when finding an indefinite integral (antiderivative). It disappears only when you have a definite integral with limits or when you apply initial/boundary conditions later.

Connected Concepts

Connected notes