242hw6solns - Math 242 Homework 6 Solutions (to the by hand...

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Unformatted text preview: Math 242 Homework 6 Solutions (to the by hand questions) Questions 1 through 5 should be done by hand. For questions 6 through 8, you can use Maple as much or as little as you like. 1. Evaluate Z sin(ln x ) dx . (May be a little tricky.) Solution. Let I = Z sin(ln x ) dx , and integrate by parts with u = sin(ln x ) and dv = 1 dx . Then du = cos(ln x ) 1 x dx and v = x . We then have I = x sin(ln x )- Z cos(ln x ) dx. Integrate by parts again, with u = cos(ln x ) and dv = 1 dx . Then du =- sin(ln x ) 1 x dx and v = x . This gives I = x sin(ln x )- x cos(ln x )- Z- sin(ln x ) dx I = x sin(ln x )- x cos(ln x )- Z sin(ln x ) dx 2 I = x sin(ln x )- x cos(ln x ) I = 1 2 x sin(ln x )- x cos(ln x ) + C. 2. Evaluate Z x x 2 + 2 dx or show that it is divergent. Solution. Consider Z M x x 2 + 2 dx . Substitute u = x 2 +2, so du = 2 x dx 1 2 du = x dx . The integral becomes 1 2 Z M 2 +2 2 1 u du = 1 2 ln( M 2 + 2)- ln 2 ....
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242hw6solns - Math 242 Homework 6 Solutions (to the by hand...

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