Chapter 4: Integration
Section 4.5: Improper Integrals
Example 4.5.2
Determine whether the improper integral ∫1∞1x ⅆx converges or diverges.
Solution
Mathematical Solution
∫1∞1x ⅆx=limt→∞∫1t1x ⅆx=limt→∞lnx1tlimt→∞lnt−ln1 = limt→∞lnt = ∞
The integral diverges.
Maple Solution
Apply Maple to the improper integral
Control-drag the integral. Context Panel: Evaluate and Display Inline
∫1∞1x ⅆx = ∞
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
∫1t1x ⅆx→assuming real rangelnt→assign to a nameq
Expression palette: Limit template Context Panel: Evaluate and Display Inline
limt→∞q = ∞
Alternate evaluation of ∫1t1x ⅆx
Append the assuming option to the integral. Context Panel: Evaluate and Display Inline
∫1t1x ⅆx assuming t>1 = lnt
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