Indefinite Integration and Standard Forms
An indefinite integral is the family of antiderivatives of a function, written with an arbitrary constant of integration.
What is Indefinite Integration and Standard Forms?
An indefinite integral is the family of antiderivatives of a function, written with an arbitrary constant of integration.
Key formula / rule: Every antiderivative differs by a constant, hence the +C.
Key points
- Compute integrals that match standard forms directly.
- Verify an antiderivative by differentiation.
Common exam trap
Omitting the constant of integration.
Definitions
- Term
Indefinite Integration and Standard Forms
- Meaning
An indefinite integral is the family of antiderivatives of a function, written with an arbitrary constant of integration.
- Term
Indefinite Integration and Standard Forms — explanation
- Meaning
Integration is recognition rather than mechanism: an integrand is matched to a standard form, or reduced to one. Building fluency with the standard list is what makes the substitution step visible.
Learning objectives
Compute integrals that match standard forms directly.
Verify an antiderivative by differentiation.
Formulae
- Key point
Every antiderivative differs by a constant, hence the +C.
- Key point
Standard forms for powers, exponentials, logarithms and trigonometric functions must be recognised on sight.
- Key point
Differentiating the answer is a complete check.
Prerequisites
MAT-U2-LC-T2-S1-C1
Common mistakes
Omitting the constant of integration.
Integrating a product by integrating each factor separately.
Keywords
Indefinite
Integration
Standard
Forms
Practice preview
Find the indefinite integral of (sec^2 x + sin x) dx.…
easy
Evaluate the indefinite integral of (tan^2 x) dx.…
medium
If d/dx [f(x)] = 4x^3 - 3x^2 + 2x - 1 and f(1) = 5, then f(x) is:…
hard