Concept of Antiderivative and Indefinite Integral and Example andTable of Antiderivatives

Antiderivative and Indefinite Integral

The concepts of antiderivative and indefinite integral are fundamental in calculus and are closely related to the process of finding the original function when its derivative is known.

Antiderivative

An antiderivative of a function f(x) is a function F(x) whose derivative is equal to f(x). In other words, if F'(x) = f(x), then F(x) is an antiderivative of f(x).

The antiderivative of a function represents the family of functions whose derivatives match the given function. It is denoted by ∫ f(x) dx.

Indefinite Integral

The indefinite integral of a function f(x) with respect to x represents the collection of all antiderivatives of f(x). It is used to find a general expression for the antiderivative of a function.

∫ f(x) dx = F(x) + C

Here, F(x) is any antiderivative of f(x), and C is the constant of integration, accounting for the fact that there could be multiple antiderivatives.

Connection to Derivatives

The concepts of antiderivative and indefinite integral are closely related to differentiation. While differentiation finds the rate of change of a function, integration finds the accumulation or reverse process. The Fundamental Theorem of Calculus connects derivatives and integrals, showing how they are inverse operations.

Antiderivative and Indefinite Integral Examples

The concepts of antiderivative and indefinite integral are fundamental in calculus and provide insights into the reverse process of differentiation and accumulation.

Antiderivative

An antiderivative of a function f(x) is a function F(x) whose derivative is equal to f(x). It is denoted by ∫ f(x) dx.

Indefinite Integral

The indefinite integral of a function f(x) represents the collection of all antiderivatives of f(x). It is denoted by ∫ f(x) dx and is given by F(x) + C, where F(x) is any antiderivative and C is the constant of integration.

Examples

Example 1

Find the antiderivative of:

∫ 3x^2 + 2x + 5 dx

Show Solution The antiderivative is: x^3 + x^2 + 5x + C

Example 2

Calculate the indefinite integral:

∫ (2sin(x) + 3cos(x)) dx

Show Solution The integral is: -2cos(x) + 3sin(x) + C

Example 3

Find the antiderivative of:

∫ (4/x) dx

Show Solution The antiderivative is: 4ln|x| + C

Example 4

Calculate the indefinite integral:

∫ (e^x + 1/x) dx

Show Solution The integral is: e^x + ln|x| + C

Example 5

Find the antiderivative of:

∫ (2x^3 + 6x^2 + 4x) dx

Show Solution The antiderivative is: (1/2)x^4 + 2x^3 + 2x^2 + C

Table of Antiderivatives

Here is an extensive table of common antiderivatives that you may encounter in calculus:

Function	Antiderivative
x^n	(1/(n+1)) * x^(n+1) + C
e^x	e^x + C
sin(x)	-cos(x) + C
cos(x)	sin(x) + C
1/x	ln|x| + C
sec^2(x)	tan(x) + C
csc(x) cot(x)	-csc(x) + C
1/(1+x^2)	arctan(x) + C
1/√(1-x^2)	arcsin(x) + C
1/(x√(x^2-1))	arcsec(x) + C
ln(x)	xln(x) - x + C
1/(x ln(x))	ln|ln(x)| + C
e^(kx)	(1/k) * e^(kx) + C
sin(kx)	-(1/k) * cos(kx) + C
cos(kx)	(1/k) * sin(kx) + C
1/(1+x)	ln|1+x| + C
1/(a^2+x^2)	(1/a) * arctan(x/a) + C
1/(√(a^2-x^2))	(1/a) * arcsin(x/a) + C
1/(a^2-x^2)	(1/(2a)) * ln|((a+x)/(a-x))| + C
ln(ax)	xln(ax) - x + C
ax^k	(a/k+1) * x^(k+1) + C

This table covers a wide range of functions and their corresponding antiderivatives. Keep in mind that these are just a few examples, and there are many more antiderivatives that you may come across in calculus.