Integration is one of the central operations in calculus because it lets you recover a quantity from its rate of change. If differentiation breaks a function into its instantaneous slope, integration builds a function back from those slopes. This is why an indefinite integral always represents a family of antiderivatives.
Basic integration rules give you fast ways to find these antiderivatives for common functions.
The symbol ∫ f(x) dx means to find a function whose derivative is f(x). The dx tells you the variable of integration, and the constant C accounts for all functions that differ only by a vertical shift. Many rules come directly from reversing familiar derivative rules for powers, exponentials, logarithms, and trigonometric functions.
In applications, integrals are used to find accumulated change, area under a curve, displacement from velocity, and total quantity from a rate.
Key Facts
- Reverse of differentiation: if F'(x) = f(x), then ∫ f(x) dx = F(x) + C.
- Power rule: ∫ x^n dx = x^(n+1)/(n+1) + C, for n ≠ -1.
- Log rule: ∫ 1/x dx = ln|x| + C.
- Constant multiple rule: ∫ k f(x) dx = k ∫ f(x) dx.
- Sum rule: ∫ [f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx.
- Common rules: ∫ e^x dx = e^x + C, ∫ cos x dx = sin x + C, ∫ sin x dx = -cos x + C.
Vocabulary
- Integral
- An integral is a mathematical operation that finds accumulated quantity or an antiderivative.
- Antiderivative
- An antiderivative of f(x) is any function F(x) whose derivative is f(x).
- Indefinite integral
- An indefinite integral gives the general family of antiderivatives and includes a constant of integration.
- Constant of integration
- The constant of integration C represents any vertical shift of an antiderivative because constants disappear when differentiated.
- Variable of integration
- The variable of integration is the variable named in the differential, such as x in dx.
Common Mistakes to Avoid
- Forgetting + C is wrong because indefinite integrals represent a family of functions, not just one function.
- Using the power rule on ∫ 1/x dx is wrong because the rule ∫ x^n dx = x^(n+1)/(n+1) + C does not work when n = -1.
- Writing ∫ sin x dx = cos x + C is wrong because the derivative of cos x is -sin x, so the correct antiderivative is -cos x + C.
- Ignoring the variable in dx is wrong because ∫ 3y dx treats y as a constant with respect to x, while ∫ 3y dy uses y as the variable.
Practice Questions
- 1 Find ∫ (6x^2 - 4x + 9) dx.
- 2 Find ∫ (2e^x + 5/x - 3cos x) dx.
- 3 A student says ∫ f'(x) dx = f(x) exactly. Explain what is missing and why it matters.