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
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Example 2
Calculate the indefinite integral:
∫ (2sin(x) + 3cos(x)) dx
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Example 3
Find the antiderivative of:
∫ (4/x) dx
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Example 4
Calculate the indefinite integral:
∫ (e^x + 1/x) dx
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Example 5
Find the antiderivative of:
∫ (2x^3 + 6x^2 + 4x) dx
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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.