Standard Integral Formulae: Complete List with Examples

Differentiation breaks a function into its rate of change; integration puts the pieces back together. An indefinite integral ∫ f(x) dx finds every function F(x) whose derivative equals f(x), and that is why every answer ends with + C, the constant of integration. Because guessing the reverse of differentiation from scratch every time would be slow, mathematicians collected the most common results into standard integral formulae — a short list every calculus student should be able to recall on sight.

These formulae matter because they are the building blocks of every harder technique. Substitution, parts, partial fractions and definite integrals all end with one of these standard forms. If you can recognise ∫ xⁿ dx instantly as xⁿ⁺¹/(n+1) + C (for n ≠ −1), you have already solved half of the integral in front of you.

Key Concepts

• Indefinite integral: ∫ f(x) dx = F(x) + C where F′(x) = f(x); C is an arbitrary constant.
• Power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, valid for every real n except n = −1.
• Exceptional case n = −1: ∫ dx/x = ln|x| + C, since no power formula works here.
• Exponentials: ∫ eˣ dx = eˣ + C and ∫ aˣ dx = aˣ/ln a + C.
• Trigonometric integrals: ∫ sin x dx = −cos x + C, ∫ cos x dx = sin x + C, ∫ sec²x dx = tan x + C, ∫ cosec²x dx = −cot x + C.
• Inverse trig forms: ∫ dx/√(1−x²) = sin⁻¹x + C, ∫ dx/(1+x²) = tan⁻¹x + C.

Formulas

• ∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx (integrate term by term)
• ∫ k·f(x) dx = k ∫ f(x) dx (constants pull out of the integral)
• ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ −1
• ∫ dx/x = ln|x| + C
• ∫ eˣ dx = eˣ + C; ∫ aˣ dx = aˣ/ln a + C
• ∫ sin x dx = −cos x + C; ∫ cos x dx = sin x + C
• ∫ dx/√(1−x²) = sin⁻¹x + C; ∫ dx/(1+x²) = tan⁻¹x + C

Worked Example

Evaluate ∫ (3x² + 2x + 5) dx. Integrate term by term using the power rule. First, ∫ 3x² dx = 3 · x³/3 = x³. Next, ∫ 2x dx = 2 · x²/2 = x². Finally, ∫ 5 dx = 5x. Add the constant of integration once at the end: x³ + x² + 5x + C. A quick check by differentiating gives 3x² + 2x + 5, which matches the integrand, confirming the answer.

Common Misconceptions

• Forgetting + C: an indefinite integral without C describes only one function, not the whole family. In school exams a missing + C costs marks.
• Using the power rule for n = −1: the integral of 1/x is a logarithm.
• Sign slips in trig integrals: ∫ sin x dx is −cos x, not +cos x. Differentiate your answer to catch these instantly.

Real-World Applications

Every physics formula obtained by integrating acceleration to get velocity, or velocity to get displacement, leans on the power rule. Engineers use these formulae to compute areas, centres of mass and moments of inertia. Economists integrate marginal-cost functions to recover total cost. In each case the standard list turns a hard reverse-differentiation puzzle into pattern matching.

Standard integral formulae are the multiplication tables of calculus. Memorise the list, respect the + C, and every advanced integration method becomes a matter of steering a difficult integral into one of these familiar shapes.