Chapter 4: Integration
Section 4.5: Improper Integrals
Example 4.5.1
Evaluate the improper integral ∫1∞1x2 ⅆx.
Solution
Mathematical Solution
∫1∞1x2 ⅆx=limt→∞∫1t1x2 ⅆx=limt→∞−1x1tlimt→∞1−1t = 1
Integrate to a finite endpoint, then in the limit, let that endpoint approach infinity.
Maple Solution
Apply Maple to the improper integral
Control-drag the given improper integral.
Context Panel: Evaluate and Display Inline
∫1∞1x2 ⅆx = 1
Integrate to a finite endpoint, then take the limit
Control-drag the integral Change the upper limit from ∞ to t
Context Panel: Simplify≻Assuming Real Range (See Figure 4.5.1(a).)
Context Panel: Assign to a Name≻q
Figure 4.5.1(a) Dialog for Real Range option
∫1t1x2 ⅆx→assuming real ranget−1t→assign to a nameq
Expression palette: Limit template Context Panel: Evaluate and Display Inline
limt→∞q = 1
Alternate evaluation of ∫1t1x2 ⅆx
Append the assuming option to the integral. Context Panel: Evaluate and Display Inline
∫1t1x2 ⅆx assuming t>1 = t−1t
The assuming option cannot be invoked through the Context Panel. It has to be appended to the end of a Maple command. The alternative is to use the Simplify≻Assuming Real Range option in the Context Panel.
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