Antiderivatives and Basic Integration

Section 1: Antiderivatives

Definition 1.1: Antiderivative

Let $f(x)$ be a function defined on an interval $I$ . A function $F(x)$ is called an antiderivative of $f(x)$ on $I$ if:

F'(x) = f(x) \quad \text{for all } x \in I

Theorem 1.1: Antiderivatives Differ by Constant

If $F(x)$ and $G(x)$ are both antiderivatives of $f(x)$ on $I$ , then:

F(x) - G(x) = C

for some constant $C$ .

Proof of Theorem 1.1:

Since $F'(x) = G'(x) = f(x)$ , we have:

(F - G)'(x) = F'(x) - G'(x) = 0

By MVT, $F(x) - G(x)$ is constant. ∎

∎

Definition 1.2: Indefinite Integral

The indefinite integral of $f(x)$ is:

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

where $F(x)$ is any antiderivative of $f(x)$ and $C$ is the constant of integration.

Section 2: Basic Integration Formulas

Power Rule

For $n \neq -1$ :

\int x^n\,dx = \frac{x^{n+1}}{n+1} + C

For $n = -1$ :

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

Exponential Functions

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

\int a^x\,dx = \frac{a^x}{\ln a} + C \quad (a > 0, a \neq 1)

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

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

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

Inverse Trigonometric Functions

\int \frac{1}{\sqrt{1-x^2}}\,dx = \arcsin x + C

\int \frac{1}{1+x^2}\,dx = \arctan x + C

Example 2.1: Basic Integration Examples

(a) $\int x^7\,dx = \frac{x^8}{8} + C$

(b) $\int e^{3x}\,dx = \frac{e^{3x}}{3} + C$

(c) $\int \cos x\,dx = \sin x + C$

(d) $\int \frac{1}{1+x^2}\,dx = \arctan x + C$

Section 3: Linearity Properties

Theorem 3.1: Linearity of Integration

\int [f(x) + g(x)]\,dx = \int f(x)\,dx + \int g(x)\,dx

\int kf(x)\,dx = k\int f(x)\,dx

Example 3.1: Linearity Application

Find: $\int (3x^2 + 2\sin x - e^x)\,dx$

\int (3x^2 + 2\sin x - e^x)\,dx = 3\int x^2\,dx + 2\int \sin x\,dx - \int e^x\,dx

= x^3 - 2\cos x - e^x + C

Section 4: Trigonometric Integrals

Example 4.1: Trigonometric Integrals

$\int \sin x\,dx = -\cos x + C$
$\int \cos x\,dx = \sin x + C$
$\int \sec^2 x\,dx = \tan x + C$
$\int \csc^2 x\,dx = -\cot x + C$

Section 5: Exponential and Logarithmic Integrals

Example 5.1: Exponential Integrals

$\int e^x\,dx = e^x + C$
$\int a^x\,dx = \frac{a^x}{\ln a} + C$
$\int \frac{1}{x}\,dx = \ln |x| + C$

Section 6: Inverse Trigonometric Integrals

Example 6.1: Inverse Trig Integrals

$\int \frac{1}{\sqrt{1-x^2}}\,dx = \arcsin x + C$
$\int \frac{1}{1+x^2}\,dx = \arctan x + C$
$\int \frac{1}{|x|\sqrt{x^2-1}}\,dx = \text{arcsec } x + C$

Section 7: Integration by Recognition

Example 7.1: Recognition Method

Find: $\int \frac{2x}{1+x^2}\,dx$

Solution: Recognize that $(1+x^2)' = 2x$ :

\int \frac{2x}{1+x^2}\,dx = \ln(1+x^2) + C

Work These Ideas by Hand — Then Check Yourself

Velocity & Acceleration Calculator — kinematics — derivatives and integrals in action
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Practice Quiz: Antiderivatives and Basic Integration

10

Questions

0

Correct

0%

Accuracy

1

If

F'(x) = f(x)

for all

x \in I

, then

F(x)

is called:

Easy

Not attempted

2

If

F(x)

and

G(x)

are both antiderivatives of

f(x)

, then:

Easy

Not attempted

3

\int x^5\,dx =

Easy

Not attempted

4

\int e^{3x}\,dx =

Easy

Not attempted

5

\int \tan x\,dx =

Medium

Not attempted

6

\int \frac{1}{1+x^2}\,dx =

Easy

Not attempted

7

\int 3^x\,dx =

Medium

Not attempted

8

\int \frac{1}{\sqrt{1-x^2}}\,dx =

Medium

Not attempted

9

\int [f(x) + g(x)]\,dx =

Medium

Not attempted

10

\int 0\,dx =

Easy

Not attempted

Frequently Asked Questions

What is the difference between antiderivative and indefinite integral?

An antiderivative F(x) is a specific function with F'(x) = f(x). The indefinite integral ∫f(x)dx is the family of all antiderivatives: F(x) + C.

Why do we add '+C' to indefinite integrals?

Because if F(x) is an antiderivative, so is F(x) + C for any constant. The '+C' represents all possible antiderivatives.

Does every function have an antiderivative?

No. But continuous functions always have antiderivatives (Fundamental Theorem). Functions with jump discontinuities may not.

How do I know which formula to use?

Identify the form of the integrand. Look for: power functions (x^n), exponentials (e^x, a^x), trig functions, inverse trig patterns (1/√(1-x²), 1/(1+x²)), and logarithms (1/x).

What is the power rule for integration?

For n ≠ -1: ∫x^n dx = x^(n+1)/(n+1) + C. For n = -1: ∫1/x dx = ln|x| + C.