The Anti-Derivative

Learning Objectives

By the end of this lesson, you will be able to:

• Understand what an anti-derivative (indefinite integral) is.

• Compute anti-derivatives of basic functions.

• Apply power rule for integration.

• Recognize and handle the constant of integration.

• Interpret anti-derivatives as the reverse process of differentiation.

Introduction: What Is an Anti-Derivative?

The anti-derivative is the reverse of differentiation.
If you know $$f^\prime(x)$$, the anti-derivative finds the original function f(x).

Formally:

If $$F^\prime(x) = f(x)$$, then F(x) is an anti-derivative of f(x).

Since differentiation removes constants, there can be many functions whose derivative is the same.
So we add a constant of integration, C.

Basic Idea

Differentiation:

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

Anti-differentiation (reverse):

$$\int 2x dx = x^{2} + C $$

That’s why integration is sometimes called “undoing the derivative.”

Notation

The symbol $$\int$$(integral sign) represents the process of finding the anti-derivative,
and $dx$ indicates the variable of integration.

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

Power Rule for Anti-Derivatives

If $$n \neq -1 $$, then

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

Examples:

1. $$\int x^{3} dx = \frac{1}{3+1} x^{3+1} +C = \frac{x^{4}}{4} + C$$

2. $$\int f^{-2} dx = \frac{1}{-2+1} x^{-2+1} +C = \frac{x^{-1}}{-1} + C = -\frac{1}{x} + C $$

natural logarithm integration

Derivative natural logarithm is $$\frac{d}{dx}ln(x) = \frac{1}{x}$$

Anti-Derivative is

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

this is very important technique for integration Trigonometry

Addition and Subtraction

$$\int f(x)+g(x)+h(x) dx = \int f(x) dx + \int g(x) dx + \int h(x) dx$$

example:

$$\int 2x^{3} -7x^{2} + 3x – 9 dx$$

$$= \int 2x^{3} dx – \int 7x^{2} dx + \int 3x dx – \int 9 dx$$

$$= 2\int x^{3} dx -7 \int x^{2} dx + 3 \int x dx – \int 9 dx$$

$$= 2\cdot\frac{1}{4} x^{4} +C -7\cdot\frac{1}{3} x^{3} + C + 3\cdot\frac{1}{2} x^{2} +C – 9x +C $$

$$= \frac{1}{2} x^{4} -\frac{7}{3} x^{3} +\frac{3}{2}x^{2}-9x+C$$

Find the equation (known initial value , find out integration constant C)

example:

$$\frac{dy}{dx} =x^{2}+4x-7$$ and the graph pass through (0,9) , find the equation.

first, Let integral $$\frac{dy}{dx}$$ : 

$$f(x)=\int x^{2}+4x-7dx = \frac{1}{3}x^{3}+4\cdot\frac{1}{2}x^{2}-7x+C = \frac{1}{3}x^{3}+2x^{2}-7x+C$$

we know $$f(x)$$ pass through (0,9). then

$$9=\frac{1}{3}\cdot0^{3}+2\cdot0^{2}-7\cdot0+C$$ therefore $$C = 9$$

$$f(x)=\frac{1}{3}x^{3}+2x^{2}-7x+9$$