Indefinite Integrals — Integration

1. What an Indefinite Integral Means

An indefinite integral is the reverse process of differentiation. It represents a whole family of functions whose derivatives match a given function. In simple personal-notes style: when we integrate, we are trying to find the original function before it was differentiated.

2. General Form of an Indefinite Integral

If the derivative of F(x) is f(x), then the indefinite integral of f(x) is F(x) plus a constant.

2.1. Definition

\int f(x) \, dx = F(x) + C

\text{where } F'(x) = f(x)

The constant C appears because differentiating any constant gives zero, so many functions share the same derivative.

3. Basic Antiderivative Patterns

Some formulas repeat so often that they become standard patterns. These patterns are the foundation for most integrals.

3.1. Power Rule for Integrals

\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{(for } n \ne -1)

3.2. Integral of 1/x

\int \frac{1}{x} \, dx = \ln|x| + C

3.3. Exponential Functions

\int e^x dx = e^x + C

\int a^x dx = \frac{a^x}{\ln a} + C

3.4. Trigonometric Functions

\int \sin x \, dx = -\cos x + C

\int \cos x \, dx = \sin x + C

\int \sec^2 x \, dx = \tan x + C

\int \csc^2 x \, dx = -\cot x + C

4. Checking the Result by Differentiation

A good way to confirm an integral is correct is to differentiate the answer. If the derivative brings you back to the original function, the integral is correct.

4.1. Example

Integrate:

\int 2x \, dx = x^2 + C

Differentiate the answer:

\frac{d}{dx}(x^2 + C) = 2x

It matches the original integrand.

5. Examples of Indefinite Integrals

These simple examples show how basic rules apply.

5.1. Example 1 — Polynomial

\int (3x^2 - 4x + 1) \, dx

= x^3 - 2x^2 + x + C

5.2. Example 2 — Exponential

\int 5e^x dx = 5e^x + C

5.3. Example 3 — Trigonometric