Standard Integrals

An antiderivative of a function f(x) is any function F(x) whose derivative equals f(x), that is, F'(x) = f(x). Because differentiation and integration are inverse operations, every differentiation rule produces a corresponding integration rule. A curated list of these pairings — called standard integrals — forms the essential toolkit every calculus student must master before tackling more complex techniques such as substitution or integration by parts. Standard integrals cover the most frequently encountered function families: power functions, exponentials, logarithms, and trigonometric functions. Each formula includes an arbitrary constant of integration, written +C, which accounts for the fact that infinitely many functions share the same derivative. For example, both x² and x² + 7 have derivative 2x, so the antiderivative of 2x is written x² + C rather than a single specific function. Memorising and fluently applying standard integrals dramatically speeds up problem solving in pure mathematics, physics, and engineering. When you encounter a definite integral — one with specific numerical limits — the constant C cancels out automatically, making the table even more practical. This article presents the key formulas, explains the reasoning behind them, and demonstrates their use through worked examples.

Key takeaways
  • The Power Rule for Integration
  • The Reciprocal and Logarithm Rule
  • Exponential Integrals
  • Trigonometric Integrals
  • Linearity of Integration
  • Verifying an Antiderivative

The Power Rule for Integration

The most fundamental standard integral comes directly from reversing the power rule of differentiation. Since d/dx [x^(n+1)/(n+1)] = x^n, we conclude that ∫x^n dx = x^(n+1)/(n+1) + C, provided n ≠ −1. For example, ∫x^3 dx = x^4/4 + C. The restriction n ≠ −1 is critical: dividing by n+1 = 0 is undefined, and that special case is handled separately by the logarithm rule.

The Reciprocal and Logarithm Rule

For the missing case n = −1, we use the fact that d/dx [ln|x|] = 1/x. Therefore ∫(1/x) dx = ln|x| + C. The absolute value is essential because ln is only defined for positive inputs, yet 1/x exists for all x ≠ 0. This rule also applies to expressions like ∫(1/(ax+b)) dx = (1/a) ln|ax+b| + C, obtained by a simple substitution.

Exponential Integrals

Because d/dx [e^x] = e^x, the exponential function is its own antiderivative: ∫e^x dx = e^x + C. For a general base: ∫a^x dx = a^x / ln(a) + C (a > 0, a ≠ 1). The more common variant involving a constant multiple in the exponent is ∫e^(kx) dx = e^(kx)/k + C, which follows from the chain rule in reverse.

Trigonometric Integrals

Reversing the standard derivatives of sine and cosine gives ∫sin(x) dx = −cos(x) + C and ∫cos(x) dx = sin(x) + C. Be careful with signs: the negative sign in the first formula surprises many students. Additional standard results include ∫sec²(x) dx = tan(x) + C, ∫cosec²(x) dx = −cot(x) + C, ∫sec(x)tan(x) dx = sec(x) + C, and ∫cosec(x)cot(x) dx = −cosec(x) + C. Each can be verified by differentiating the right-hand side.

Linearity of Integration

Integration is linear: ∫[af(x) + bg(x)] dx = a∫f(x) dx + b∫g(x) dx, where a and b are constants. This means you can split sums and pull out constant multipliers before applying individual standard formulas. For instance, ∫(3x² + 5cos x) dx = 3·(x³/3) + 5·sin(x) + C = x³ + 5sin(x) + C. Linearity is what makes the standard integral table usable for polynomials of any degree.

Verifying an Antiderivative

You can always check an antiderivative by differentiating your answer. If d/dx[F(x)] = f(x), then F(x) is correct. This verification step is a powerful self-checking tool: it requires only differentiation (which most students find easier) and catches errors such as wrong signs or missing factors. For example, to check ∫cos(3x) dx = sin(3x)/3 + C, differentiate: d/dx[sin(3x)/3] = 3cos(3x)/3 = cos(3x) ✓.

The Key Equation

∫xn dx=xn+1n+1+C(n≠−1)\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)
  • x — Variable of integration (dimensionless): The independent variable being integrated with respect to.
  • n — Exponent (dimensionless): Any real number except −1; determines the power of x in the integrand.
  • C — Constant of integration (dimensionless): An arbitrary real constant representing the family of all antiderivatives.
  • n+1 — Incremented exponent (dimensionless): The new power after integration, which also appears in the denominator as the divisor.

Worked Example

Problem. Find the antiderivative of f(x) = 4x³ − 6x + 1/x + 2e^(3x), and verify one term by differentiation.

F(x)=x4−3x2+ln⁡∣x∣+23e3x+CF(x) = x^4 - 3x^2 + \ln|x| + \tfrac{2}{3}e^{3x} + C

Answer. F(x) = x⁴ − 3x² + ln|x| + (2/3)e^(3x) + C

Figure
Diagram
Standard Integrals Reference TableIntegrand f(x)Antiderivative F(x) + Cxⁿ (n ≠ −1)xⁿ⁺¹ / (n+1) + C1 / xln|x| + Ceˣeˣ + Ce^(kx)e^(kx) / k + Csin(x)−cos(x) + Ccos(x)sin(x) + Csec²(x)tan(x) + Ccosec²(x)−cot(x) + Caˣ (a > 0, a ≠ 1)aˣ / ln(a) + CAll results can be verified by differentiating the right-hand side.
Origins · 1 minGottfried Wilhelm Leibniz and how this was discovered
GW
Gottfried Wilhelm Leibniz
1646–1716 · German

Leibniz independently developed calculus in the 1670s, introducing the integral sign ∫ (an elongated S for 'summa') and the notation dx that we still use today. His systematic approach to antidifferentiation and his publication of calculus rules in 1684–1686 established the formal framework from which standard integral tables directly descend. Leibniz was also a philosopher, diplomat, and inventor. He designed a mechanical calculator called the Stepped Reckoner that could multiply and divide — a remarkable feat of engineering in the 17th century.