Antiderivatives and Indefinite Integrals
The Antiderivative
Previously, we learned that the derivative of a function represents the rate of change of a function and we determined different methods in computing derivatives. Suppose we have a function [latex]f(x) = x^2[/latex], as we know, the derivative of the function is [latex]f'(x) = 2x[/latex]
When we think of integrals and integration, we can think of it as the ‘reverse’ operation to differentiation. First, let’s discuss the definition of the antiderivative. Well, as hinted in the name, an antiderivative is the opposite of the derivative. Take a look at the following definitions:
Note:
Given a function [latex]f(x)[/latex], the antiderivative of [latex]f(x)[/latex] is any function [latex]F(x)[/latex] such that the derivative of [latex]F(x)[/latex] is [latex]f(x)[/latex]
Indefinite Integral:
[latex]\int{f(x)dx} = F(x) + C[/latex]
In the above definition, the [latex]\int[/latex] symbol represents integration and f(x) is the function to be integrated, known as the integrand. C represents an arbitrary constant, which of course can be any constant value.
So, using this definition, we can say that:
[latex]\int 2xdx = x^2 + C[/latex].
Notice that C could be any constant, and no matter what constant it is, the derivative of [latex]x^2+C[/latex] will be [latex]2x[/latex]
Integral Power Rule
Let us further explore the concept of antiderivative and the indefinite integral. Suppose you are given a function which is:
[latex]f(x) = 5x^4[/latex]
Let’s see how we would evaluate [latex]\int 5x^4 dx[/latex]
So first, we are being asked to find the indefinite integral of the function, which is the same as finding the most general antiderivative of the function.
If you remember from the derivatives section, we learned about the derivative power rule to find the derivative of polynomial functions. To find the antiderivative of polynomial functions, we will use the integral power rule, which is basically the opposite of the derivative power rule.
The power rule for integration is defined as follows:
Integral Power Rule:
[latex]\int x^n dx = \frac{x^{n+1}}{n+1} + C[/latex]
Applying this rule to the function given, we find the integral as follows:
[latex]\large{\int 5x^4 dx = \frac{5(x^{4+1})}{4+1} = \frac{5x^5}{5} = x^5 + C }[/latex]
Remember to always add the constant C when evaluating indefinite integrals.
Example
Find the general antiderivative of the following function:
[latex]\large{f(x) = e^x}[/latex]
From our knowledge of the derivative of exponential functions, we know that the derivative of [latex]\large{e^x}[/latex] is [latex]\large{e^x}[/latex].
Thus, we can conclude that the antiderivative must be the same. Hence, we have:
[latex]\large{\int{e^x dx} = e^x + C}[/latex].
Remember to add the constant C at the end of the indefinite integral.
Example
Evaluate the following indefinite integral:
[latex]\large{\int \sqrt{x} dx}[/latex]
Step 1:
First, we can rewrite the integral as shown:
[latex]\large{\int{\sqrt{x}dx} = \int{x^{\frac{1}{2}}dx}}[/latex]
Step 2:
Now, we can simply apply the integral power rule, as shown:
[latex]\large{\int{x^{\frac{1}{2}}dx} = \frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1} + C}[/latex]
Simplifying, we have:
[latex]\large{= \frac{x^{\frac{3}{2}}}{\frac{3}{2}} + C}[/latex]
[latex]\large{= \frac{2}{3}x^{\frac{3}{2}} + C}[/latex]
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