Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative. take C=0. x(x21)3dx
The given indefinite integral .
From the graph, it is reasonable to say is the antiderivative of .
i) The substitution rule
If is a differentiable function whose range is I and f is continuous on I, then . ii) Indefinite integral
The given integral is
Here, use the substitution method.
Differentiating with respect to .
Therefore, the given integral becomes
Using concept ii),
Hence, the solution is
Now, take and sketch the graph of the function and its antiderivative
The blue graph is for and the red graph is for
The answer is reasonable since where is positive, there is increasing; where is negative, there is decreasing; where changes from negative to positive, there has a local minimum; and where changes from positive to negative, there has a local maximum.
From the graph, it is reasonable to say is the antiderivative of