Department of Mathematics
Name: _______________________________ Improper Integrals – Worksheet 1Definition: The integral abfx dx is called improper if… (1)(2)Major Note: Improper integrals are treated as limits. (1) The integral a∞fx dx is treated as… (2) The integral -∞bfx dx is treated as… (3) The integral -∞∞fx dx is treated as…(4) If c is a point where an infinite discontinuity of f occurs inside a,b then the integral abfx dx is treated as… Example 1a: An amazing example with a finite amount of area in an infinite region!Evaluate 011x dxComment: The above example means that sometimes with a FINITE amount of paint you can (theoretically) paint an infinitely long region (albeit this region gets “real small near infinity”). Example 1b: A non-amazing example.Evaluate 1∞1x dxComment: The above example is an infinite region that would take an infinite amount of paint to paint. Example 2: Evaluate 1∞1x dxExample 3: Evaluate 1∞1x3 dx-38100210185 Notice that in examples 1b, 2, and 3, the integral is of the form 1∞1xp dx, where p is some real number. Fill in the blanks with either converges or diverges.When p>1, the integral1∞1xp dx _______________________.When p≤1, the integral 1∞1xp dx _______________________.00 Notice that in examples 1b, 2, and 3, the integral is of the form 1∞1xp dx, where p is some real number. Fill in the blanks with either converges or diverges.When p>1, the integral1∞1xp dx _______________________.When p≤1, the integral 1∞1xp dx _______________________.Example 3: Evaluate -∞∞ex1+ex dxExample 4: Evaluate 0∞3x+1e-x dxExample 5: Evaluate 031x-12 dxNotice that we obtain the wrong answer if we do not realize this is improper!!!Example 6: Evaluate 031x-123 dxExample 7: Evaluate -∞∞1x2+1 dx (Remember - symmetry helps make things easier!!) ................
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