The value of the indefinite integral function .
The value of the indefinite integral is
The integral function is .
Consider as u.
Differentiate both sides of the Equation (1).
Rearrange Equation (2) to find the value of as shown below.
Substitute u for , for , and for in the function as shown below.
The expression to find the indefinite integral value using Equation (4) as shown below.
Substitute for u in Equation (5) as shown below.
Therefore, the value of the indefinite integral is .
Apply the Theorem:
The integral function is traditionally used for an antiderivative of f and is called an indefinite integral.
Substitute for and for in Equation (6) as shown below.
Draw the graph for the function and its antiderivative using Equation (7) as shown below:
Refer to Figure 1.
The function Increases with positive value of and decreases with negative value of .
The minimum value of the function occurs at . Hence, it is reasonable from the graphical evidence.
Therefore, is an antiderivative of .
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